🧠:[<think>Okay, let's try to figure out this problem. So, we have the Wolf and Ivan the Prince 20 versts away from the source of living water. The Wolf is carrying Ivan there at a speed of 3 versts per hour. To revive Ivan, they need 1 liter of water. The source produces half a liter per hour. There's a Crow at the source that can carry as much water as needed, but when it flies towards the Wolf and Ivan, it flies at 6 versts per hour and spills a quarter liter of water each hour. We need to find out after how many hours Ivan can be revived.First, let me break down the problem. The Wolf is moving towards the source at 3 versts per hour, so the distance between them decreases over time. Meanwhile, the Crow is collecting water at the source, which produces 0.5 liters per hour. The Crow needs to collect some amount of water and then fly to meet the Wolf and Ivan. However, while flying, it spills 0.25 liters per hour. So the amount of water the Crow can deliver depends on how long it takes to reach the Wolf and Ivan.The key here is to determine when the Crow should leave the source so that by the time it reaches the Wolf, it has enough water (1 liter) to revive Ivan. We need to consider both the time it takes for the Crow to collect the water and the time it takes to fly to the Wolf, while also accounting for the water spilled during the flight.Let me start by calculating how the distance between the Wolf and the source changes over time. The Wolf is moving towards the source at 3 versts per hour, starting from 20 versts away. So, the distance remaining at time t hours would be 20 - 3t versts. However, since the Wolf is moving towards the source, the distance decreases, so if the Crow starts flying at time t, the distance it needs to cover to meet the Wolf is 20 - 3t versts. But wait, actually, the Crow is starting at the source, so when the Crow departs, the Wolf is at some distance. Hmm, maybe I need to model this differently.Wait, actually, time is passing while the Crow collects water. Let's say the Crow starts collecting water at time 0. If the Crow collects water for 'x' hours, then it has 0.5x liters of water. Then it departs towards the Wolf. At that time, the Wolf has been traveling for 'x' hours towards the source, so the Wolf's position is 20 - 3x versts away from the source. Then, the Crow needs to fly from the source to meet the Wolf. The Crow's speed is 6 versts per hour, and the Wolf is moving towards the source at 3 versts per hour. So their closing speed is 6 + 3 = 9 versts per hour. The distance between them when the Crow departs is 20 - 3x versts. Therefore, the time it takes for the Crow to reach the Wolf is (20 - 3x)/9 hours.During this flight time, the Crow is spilling water at 0.25 liters per hour. So the total water spilled is 0.25 * (time of flight) = 0.25 * (20 - 3x)/9 liters. The amount of water the Crow carries is 0.5x liters. Therefore, the water delivered to the Wolf is 0.5x - 0.25*(20 - 3x)/9 liters. We need this to be at least 1 liter.So, setting up the equation:0.5x - (0.25*(20 - 3x)/9) ≥ 1Let's solve for x.First, multiply both sides by 9 to eliminate the denominator:0.5x*9 - 0.25*(20 - 3x) ≥ 9*1Which simplifies to:4.5x - 5 + 0.75x ≥ 9Combine like terms:(4.5x + 0.75x) -5 ≥95.25x -5 ≥9Add 5 to both sides:5.25x ≥14Divide both sides by 5.25:x ≥14 /5.25Calculating 14 divided by 5.25:5.25 goes into 14 twice (5.25*2=10.5), remainder 3.53.5 /5.25 = 2/3So x ≥2 + 2/3 = 2.666... hours. So x is at least 8/3 hours (since 2/3 is approximately 0.666...)But x is the time the Crow spends collecting water before departing. Then, after departing, the time taken to reach the Wolf is (20 -3x)/9 hours. So total time until revival is x + (20 -3x)/9 hours.We need to compute this total time.But wait, let's check. The problem says "after how many hours will it be possible to revive Ivan the Prince?" So the total time is x (collecting time) plus flight time. Let's substitute x = 8/3 hours into the flight time:Flight time = (20 -3*(8/3))/9 = (20 -8)/9 =12/9 =4/3 hours. So total time is 8/3 +4/3 =12/3=4 hours.Therefore, the answer is 4 hours. Hmm, that matches the given answer. Let me verify if the steps are correct.Wait, but let's check when x=8/3, which is 2 hours and 40 minutes. At that time, the Wolf has traveled 3*(8/3)=8 versts, so remaining distance is 20-8=12 versts. The Crow flies at 6 versts per hour towards the Wolf, which is moving at 3 versts per hour towards the source. So their closing speed is 9 versts per hour. So time to meet is 12/9 =4/3 hours, which is 1 hour and 20 minutes. Total time is 2h40m +1h20m=4 hours. Correct.Also, the amount of water collected is 0.5*(8/3)=4/3 liters≈1.333 liters. The flight time is 4/3 hours, so spilled water is 0.25*(4/3)=1/3 liters. Therefore, delivered water is 4/3 -1/3=1 liter. Exactly what's needed. So that works.Therefore, the answer is 4 hours.Final AnswerThe time after which Ivan the Prince can be revived is boxed{4} hours.</think>The Wolf and Ivan the Prince are 20 versts away from the source of living water, moving towards it at 3 versts per hour. The Crow collects water at the source, which produces 0.5 liters per hour, and then flies to meet the Wolf, spilling 0.25 liters per hour. The Crow flies at 6 versts per hour. We need to determine the total time required to collect and deliver 1 liter of water.1. Distance Calculation: - Let ( x ) be the time (in hours) the Crow spends collecting water. - After ( x ) hours, the Wolf has traveled ( 3x ) versts, reducing the distance to the source to ( 20 - 3x ) versts.2. Flight Time Calculation: - The Crow flies towards the Wolf at 6 versts per hour, while the Wolf moves towards the source at 3 versts per hour. Their closing speed is ( 6 + 3 = 9 ) versts per hour. - The time for the Crow to meet the Wolf is ( frac{20 - 3x}{9} ) hours.3. Water Collection and Spillage: - Water collected by the Crow: ( 0.5x ) liters. - Water spilled during flight: ( 0.25 times frac{20 - 3x}{9} ) liters. - Water delivered: ( 0.5x - 0.25 times frac{20 - 3x}{9} geq 1 ).4. Solving the Inequality: - Multiply by 9: ( 4.5x - 0.25(20 - 3x) geq 9 ) - Simplify: ( 4.5x - 5 + 0.75x geq 9 ) - Combine terms: ( 5.25x - 5 geq 9 ) - Solve for ( x ): ( 5.25x geq 14 ) ⟹ ( x geq frac{14}{5.25} = frac{8}{3} ) hours (2 hours 40 minutes).5. Total Time Calculation: - Flight time: ( frac{20 - 3 times frac{8}{3}}{9} = frac{12}{9} = frac{4}{3} ) hours (1 hour 20 minutes). - Total time: ( frac{8}{3} + frac{4}{3} = 4 ) hours.Thus, the time after which Ivan the Prince can be revived is boxed{4} hours.,<think>Okay, let me try to work through this problem step by step. Hmm, so we have Ivan the Prince and the Wolf who are 20 versts away from the source of living water. The Wolf is carrying Ivan towards the source at a speed of 3 versts per hour. To revive Ivan, they need 1 liter of water. The source produces water at a rate of half a liter per hour. There's also a Crow at the source with unlimited carrying capacity, which can fly at 6 versts per hour. But when the Crow flies, it spills a quarter liter of water each hour. The question is, after how many hours will it be possible to revive Ivan the Prince?Alright, let's break this down. First, I need to figure out how the Crow can get the required 1 liter of water to Ivan before the Wolf and Ivan reach the source. But wait, the Wolf is moving towards the source, right? So the distance between them and the source is decreasing over time. At the same time, the Crow needs to collect water from the source, which is being produced at 0.5 liters per hour, and then fly to meet them, spilling water along the way.I think the key here is to determine the optimal time for the Crow to start flying so that when it meets Ivan and the Wolf, it has at least 1 liter of water left, considering both the production rate and the spillage during flight.Let's denote the time variable. Let’s say t is the number of hours after the Crow starts flying. Wait, but when does the Crow start collecting water? The problem says the Crow must collect the water first, then fly towards them. So maybe the Crow starts collecting water at time 0, and after some time T, it has collected enough water, then flies towards them, taking some additional time. But we need to figure out the total time until the Crow delivers the water.But maybe another approach: The Crow can collect water while the Wolf is moving towards the source. The Crow can start flying at a certain time when it has enough water, and then calculate how much time it takes to reach Ivan and the Wolf, considering the spillage during flight. The total time would be the time spent collecting water plus the time flying. But we need to ensure that when the Crow arrives, Ivan is still alive, i.e., hasn't reached the source yet.Wait, actually, the Wolf is moving towards the source, so Ivan and the Wolf will reach the source in 20 versts / 3 versts per hour ≈ 6.666... hours. So the Crow has to deliver the water before 6 and 2/3 hours have passed.But maybe the Crow can collect water for some time, then fly to meet them. The Crow can start collecting water at time 0, then after t1 hours, it has collected 0.5 * t1 liters. Then, it flies towards them. The time it takes to fly to meet them depends on how far they are from the source at that time.But during the flight, the Crow is spilling water at 0.25 liters per hour. So if the flight takes t2 hours, the amount of water remaining is 0.5 * t1 - 0.25 * t2. We need this to be at least 1 liter. Also, the distance the Crow has to cover is 20 - 3 * (t1 + t2) versts, because while the Crow was collecting water for t1 hours, the Wolf was moving towards the source, covering 3*t1 versts, and then during the flight time t2, the Wolf covers another 3*t2 versts. So the remaining distance when the Crow starts flying is 20 - 3*t1, and during flight, the Wolf moves another 3*t2, so the total distance when the Crow meets them is 20 - 3*(t1 + t2). But the Crow is flying at 6 versts per hour towards them, so the distance the Crow covers is 6*t2. But the remaining distance when the Crow starts flying is 20 - 3*t1, and during flight, the Wolf is moving closer, so the distance decreases by 3*t2. So the Crow has to cover the distance 20 - 3*t1 - 3*t2 = 6*t2. Wait, is that correct?Wait, the total distance the Crow needs to fly is equal to the distance between the source and Ivan/Wolf at the time the Crow starts flying, which is 20 - 3*t1, minus the distance the Wolf covers during the Crow's flight time, which is 3*t2. So the distance the Crow flies is 6*t2 = (20 - 3*t1) - 3*t2. Let me write that equation:6*t2 = (20 - 3*t1) - 3*t2So let's solve for t2 in terms of t1:6t2 + 3t2 = 20 - 3t19t2 = 20 - 3t1t2 = (20 - 3t1)/9Okay, so the flight time t2 is (20 - 3t1)/9 hours.Now, the amount of water the Crow has when it starts flying is 0.5*t1 liters. During the flight, it spills 0.25 liters per hour, so total spillage is 0.25*t2 liters. So the remaining water is 0.5*t1 - 0.25*t2. We need this to be at least 1 liter:0.5*t1 - 0.25*t2 ≥ 1But t2 is (20 - 3t1)/9, so substitute that in:0.5*t1 - 0.25*(20 - 3t1)/9 ≥ 1Let me compute this step by step.First, calculate 0.25*(20 - 3t1)/9:0.25*(20 - 3t1)/9 = (5 - 0.75t1)/9 ≈ 5/9 - (0.75/9)t1 ≈ 0.555... - 0.0833...t1So the inequality becomes:0.5*t1 - (0.555... - 0.0833...t1) ≥ 1Let me convert decimals to fractions for accuracy.0.5 is 1/2, 0.25 is 1/4, 20 is 20, 3t1 is 3t1.So original inequality:(1/2)t1 - (1/4)*[(20 - 3t1)/9] ≥ 1Let's compute the second term:(1/4)*(20 - 3t1)/9 = (20 - 3t1)/(36)So the inequality is:(1/2)t1 - (20 - 3t1)/36 ≥ 1Multiply both sides by 36 to eliminate denominators:18t1 - (20 - 3t1) ≥ 36Simplify:18t1 -20 + 3t1 ≥ 36Combine like terms:21t1 -20 ≥3621t1 ≥56t1 ≥56/21 ≈2.666... hours ≈2 hours and 40 minutes.So t1 must be at least 56/21 hours, which is 8/3 hours ≈2.6667 hours.But t2 is (20 -3t1)/9. So plugging t1 =56/21=8/3≈2.6667:t2=(20 -3*(8/3))/9=(20 -8)/9=12/9=4/3≈1.3333 hours≈1 hour 20 minutes.So total time is t1 + t2=8/3 +4/3=12/3=4 hours.So the total time is 4 hours.Wait, so this suggests that if the Crow collects water for 8/3 hours (2h40m), then flies for 4/3 hours (1h20m), delivering the water at the 4-hour mark, which is before the Wolf and Ivan reach the source at 20/3≈6.6667 hours. Therefore, it's possible.But let's verify the amount of water. When the Crow starts flying, it has collected 0.5*t1=0.5*(8/3)=4/3≈1.3333 liters. During the flight time of 4/3 hours, it spills 0.25*4/3≈0.3333 liters. So remaining water is 4/3 -1/3=1 liter exactly. Perfect, that's exactly the amount needed.Therefore, the total time is 4 hours.I think that's the answer. Let me just check if there's another possible way, like maybe multiple trips? But the problem says the Crow has unlimited carrying capacity, so it can carry all the water at once. Therefore, it just needs to collect enough water, fly once, and deliver it. So the calculation above seems correct.Therefore, the answer is 4 hours.Final AnswerThe number of hours after which it will be possible to revive Ivan the Prince is boxed{4}.</think>The Wolf and Ivan the Prince are 20 versts away from the source of living water, moving towards it at 3 versts per hour. The Crow needs to collect 1 liter of water, which is produced at 0.5 liters per hour, and fly to meet them, spilling 0.25 liters per hour. To determine the total time required, we need to calculate the time the Crow spends collecting water ((t_1)) and flying ((t_2)). The Crow must collect enough water and fly to meet them while they are still en route.1. Distance Calculation: - The Wolf reduces the distance to the source by 3 versts per hour. - The Crow flies at 6 versts per hour. When the Crow starts flying after (t_1) hours, the remaining distance is (20 - 3t_1). During the flight time (t_2), the Wolf covers another (3t_2) versts. - The equation for the distance the Crow flies is (6t_2 = (20 - 3t_1) - 3t_2), leading to (t_2 = frac{20 - 3t_1}{9}).2. Water Collection and Spillage: - The Crow collects (0.5t_1) liters of water. - During flight, it spills (0.25t_2) liters. The remaining water must be at least 1 liter: [ 0.5t_1 - 0.25 left(frac{20 - 3t_1}{9}right) geq 1 ] - Solving this inequality: [ frac{1}{2}t_1 - frac{20 - 3t_1}{36} geq 1 ] Multiplying by 36: [ 18t_1 - (20 - 3t_1) geq 36 implies 21t_1 geq 56 implies t_1 geq frac{56}{21} = frac{8}{3} text{ hours} ]3. Flight Time Calculation: - Substituting (t_1 = frac{8}{3}) into (t_2): [ t_2 = frac{20 - 3 left(frac{8}{3}right)}{9} = frac{12}{9} = frac{4}{3} text{ hours} ]4. Total Time: - Total time (t_1 + t_2 = frac{8}{3} + frac{4}{3} = 4) hours.Thus, the number of hours after which it will be possible to revive Ivan the Prince is (boxed{4}).]

🔑:# 5. Task 5The Wolf and Ivan the Prince are 20 versts away from the source of living water, and the Wolf is carrying Ivan the Prince there at a speed of 3 versts per hour. To revive Ivan the Prince, one liter of water is required, which flows from the source at a rate of half a liter per hour. At the source, there is a Crow with unlimited carrying capacity, which must collect the water, after which it will fly towards the Wolf and Ivan the Prince, flying 6 versts per hour and spilling a quarter liter of water each hour. After how many hours will it be possible to revive Ivan the Prince? Points for the task: 8.Answer: 4

🧠:[<think>Okay, so I need to find the range of the real number ( a ) such that the function ( f(x) = frac{a - sin x}{cos x} ) is monotonically increasing on the interval ( left( frac{pi}{6}, frac{pi}{3} right) ). Hmm, let's break this down step by step. First, if a function is monotonically increasing on an interval, its derivative should be non-negative throughout that interval. So my plan is to compute the derivative of ( f(x) ), set it to be greater than or equal to zero, and solve for ( a ). That should give me the required range. Let me start by finding the derivative of ( f(x) ).The function is a quotient, so I need to apply the quotient rule. The quotient rule states that if you have ( f(x) = frac{u(x)}{v(x)} ), then the derivative ( f'(x) ) is ( frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ). Here, ( u(x) = a - sin x ) and ( v(x) = cos x ). Let's compute the derivatives of ( u ) and ( v ):- ( u'(x) = -cos x ) (since the derivative of ( sin x ) is ( cos x ), and there's a negative sign)- ( v'(x) = -sin x ) (the derivative of ( cos x ) is ( -sin x ))Now, applying the quotient rule:( f'(x) = frac{(-cos x)(cos x) - (a - sin x)(-sin x)}{cos^2 x} )Let me simplify the numerator step by step. First, compute each part:1. ( (-cos x)(cos x) = -cos^2 x )2. ( (a - sin x)(-sin x) = -a sin x + sin^2 x ) - But since there's a negative sign in front of this term in the numerator, it becomes ( +a sin x - sin^2 x )So combining these two parts, the numerator becomes:( -cos^2 x + a sin x - sin^2 x )Let me write that out:Numerator = ( -cos^2 x - sin^2 x + a sin x )Wait, hold on, is that correct? Let me double-check. The first term is ( -cos^2 x ), the second term is ( +a sin x ), and the third term is ( -sin^2 x ). So combining the ( -cos^2 x ) and ( -sin^2 x ), that's ( -(cos^2 x + sin^2 x) ). But ( cos^2 x + sin^2 x = 1 ), so that simplifies to ( -1 ). Therefore, the numerator simplifies to:( -1 + a sin x )So putting it all together, the derivative ( f'(x) ) is:( f'(x) = frac{-1 + a sin x}{cos^2 x} )Now, since we want ( f(x) ) to be monotonically increasing on ( left( frac{pi}{6}, frac{pi}{3} right) ), the derivative ( f'(x) ) must be greater than or equal to zero on that interval. The denominator ( cos^2 x ) is always positive on this interval because ( cos x ) is positive in the first quadrant (from ( 0 ) to ( pi/2 )), and here we're between ( pi/6 ) and ( pi/3 ), which is within ( 0 ) to ( pi/2 ). So the sign of ( f'(x) ) is determined solely by the numerator ( -1 + a sin x ).Therefore, the condition ( f'(x) geq 0 ) is equivalent to:( -1 + a sin x geq 0 )( Rightarrow a sin x geq 1 )( Rightarrow a geq frac{1}{sin x} )But this inequality must hold for all ( x ) in ( left( frac{pi}{6}, frac{pi}{3} right) ). Therefore, ( a ) must be greater than or equal to the maximum value of ( frac{1}{sin x} ) on this interval.So now I need to find the maximum of ( frac{1}{sin x} ) on ( left( frac{pi}{6}, frac{pi}{3} right) ). Let's analyze ( sin x ) on this interval.The function ( sin x ) is increasing on ( [0, pi/2] ), so on the interval ( [pi/6, pi/3] ), it is also increasing. Therefore, the minimum value of ( sin x ) on this interval is ( sin (pi/6) = 1/2 ), and the maximum value is ( sin (pi/3) = sqrt{3}/2 ).Therefore, ( frac{1}{sin x} ) is decreasing on this interval because ( sin x ) is increasing. So the maximum of ( frac{1}{sin x} ) occurs at the left endpoint ( x = pi/6 ), which is ( 1/(1/2) = 2 ), and the minimum occurs at the right endpoint ( x = pi/3 ), which is ( 1/(sqrt{3}/2) = 2/sqrt{3} approx 1.1547 ).Therefore, to satisfy ( a geq frac{1}{sin x} ) for all ( x ) in ( (pi/6, pi/3) ), ( a ) must be at least as large as the maximum of ( frac{1}{sin x} ), which is 2. However, we have to be careful here. The interval is open, ( (pi/6, pi/3) ), not including the endpoints. But since ( sin x ) is continuous and increasing, even on the open interval, the supremum of ( frac{1}{sin x} ) is still 2 (approached as ( x ) approaches ( pi/6 ) from the right). Therefore, to ensure that ( a geq frac{1}{sin x} ) for all ( x ) in the open interval, ( a ) must be greater than or equal to 2. If ( a = 2 ), then at ( x ) approaching ( pi/6 ), ( frac{1}{sin x} ) approaches 2, so ( a = 2 ) would make ( a sin x ) approach 2*(1/2) = 1, so ( -1 + a sin x ) approaches 0. But since the interval is open, ( x ) never actually reaches ( pi/6 ), so ( sin x ) is always greater than 1/2 on ( (pi/6, pi/3) ). Wait, actually, hold on. If ( x ) is in ( (pi/6, pi/3) ), then ( sin x ) is in ( (1/2, sqrt{3}/2) ). Therefore, ( frac{1}{sin x} ) is in ( (2/sqrt{3}, 2) ). Therefore, the supremum of ( frac{1}{sin x} ) on the open interval is 2, but it's never actually achieved. So if ( a = 2 ), then ( a sin x ) would be 2 sin x. The minimal value of ( a sin x ) when ( a = 2 ) would be 2*(1/2) = 1, but since ( x ) is greater than ( pi/6 ), ( sin x > 1/2 ), so ( 2 sin x > 1 ). Therefore, ( -1 + 2 sin x > 0 ) for all ( x ) in ( (pi/6, pi/3) ). Wait, that seems contradictory to my previous thought. Let me check.If ( a = 2 ), then ( -1 + a sin x = -1 + 2 sin x ). Let's evaluate this at ( x ) approaching ( pi/6 ) from the right. Then ( sin x ) approaches ( 1/2 ), so ( -1 + 2*(1/2) = -1 + 1 = 0 ). But since ( x ) is approaching ( pi/6 ) from the right, in the open interval ( (pi/6, pi/3) ), ( sin x ) is strictly greater than ( 1/2 ), so ( -1 + 2 sin x > 0 ). Therefore, even when ( a = 2 ), the derivative ( f'(x) ) is strictly positive throughout the interval. However, if ( a < 2 ), then near ( x = pi/6 ), ( sin x ) is close to ( 1/2 ), so ( a sin x ) would be less than 1, making ( -1 + a sin x ) negative. Therefore, ( a must be at least 2 to ensure that ( -1 + a sin x geq 0 ) for all ( x ) in ( (pi/6, pi/3) ). But wait, if we take ( a = 2 ), then at ( x ) approaching ( pi/6 ), ( -1 + 2 sin x ) approaches 0. But since ( x ) never actually equals ( pi/6 ), maybe even ( a = 2 ) is sufficient? Wait, but actually, since the interval is open, ( x ) can get arbitrarily close to ( pi/6 ). So if ( a = 2 ), then for any ( x ) in ( (pi/6, pi/3) ), ( sin x > 1/2 ), so ( 2 sin x > 1 ), so ( -1 + 2 sin x > 0 ). Therefore, even though the limit as ( x ) approaches ( pi/6 ) from the right is 0, the derivative never actually reaches zero in the open interval. Therefore, ( a = 2 ) is acceptable because ( f'(x) > 0 ) for all ( x ) in the open interval. However, if ( a ) were less than 2, say ( a = 1.9 ), then near ( x = pi/6 ), ( sin x ) is just over 0.5, so ( 1.9 * 0.5 = 0.95 ), and ( -1 + 0.95 = -0.05 < 0 ), which would make the derivative negative. Hence, ( a must be at least 2 to ensure the derivative is non-negative throughout the interval. Similarly, if we consider the upper end of the interval ( x = pi/3 ), ( sin x = sqrt{3}/2 approx 0.866 ), so ( a sin x ) would be ( a * 0.866 ). For this to be greater than or equal to 1, ( a geq 1 / (sqrt{3}/2) = 2 / sqrt{3} approx 1.1547 ). But since we need the inequality to hold for all ( x ) in the interval, the most restrictive condition is the lower bound of ( a geq 2 ). Wait, this seems conflicting. Let me verify.Wait, perhaps I made a mistake here. Let's re-examine.We have the condition ( a geq frac{1}{sin x} ) for all ( x ) in ( (pi/6, pi/3) ). The function ( frac{1}{sin x} ) on this interval is decreasing because ( sin x ) is increasing. Therefore, the maximum value of ( frac{1}{sin x} ) occurs at the left endpoint (as ( x ) approaches ( pi/6 )), which is 2, and the minimum value is at the right endpoint (( x = pi/3 )), which is ( 2 / sqrt{3} ). Therefore, to satisfy ( a geq frac{1}{sin x} ) for all ( x ) in the interval, ( a ) must be greater than or equal to the maximum value of ( frac{1}{sin x} ), which is 2. If ( a geq 2 ), then ( a sin x geq 2 sin x geq 2 * (1/2) = 1 ), so ( -1 + a sin x geq 0 ). Therefore, the derivative is non-negative.But here's the confusion: when we take ( a = 2 ), the derivative at ( x ) approaching ( pi/6 ) is approaching zero. However, since ( x ) is never actually ( pi/6 ) (it's an open interval), the derivative is always positive. Hence, the function is still monotonically increasing because the derivative does not become negative anywhere in the interval. Therefore, ( a geq 2 ) is the correct range.Wait, but let's think again. Suppose ( a = 2 ). Then ( f'(x) = frac{-1 + 2 sin x}{cos^2 x} ). Since ( x ) is in ( (pi/6, pi/3) ), ( sin x ) is in ( (1/2, sqrt{3}/2) ). Therefore, ( 2 sin x ) is in ( (1, sqrt{3}) ). So ( -1 + 2 sin x ) is in ( (0, sqrt{3} -1) ), which is positive. Therefore, even when ( a = 2 ), the derivative is positive throughout the interval. So ( a = 2 ) is acceptable. If ( a > 2 ), then ( -1 + a sin x ) would be even larger, so the derivative is positive. If ( a < 2 ), then near ( x = pi/6 ), ( sin x ) is close to 1/2, so ( a sin x ) is less than 1, leading to ( -1 + a sin x < 0 ), which would make the derivative negative. Therefore, the minimal value of ( a ) is 2. There is no upper bound on ( a ), right? Wait, but the problem says "the range of the real number a", so maybe only a lower bound? Let me check.If ( a ) is larger than 2, say 3, then ( -1 + 3 sin x ) would be even more positive, so derivative is positive. Hence, the function would still be increasing. Therefore, as ( a ) increases beyond 2, the derivative becomes more positive. Therefore, the range of ( a ) is ( [2, infty) ). But wait, the question says "the range of the real number a", so maybe it's only requiring a lower bound? Let me confirm.But hold on, let's verify with an example. Let's take ( a = 3 ). Then ( f(x) = (3 - sin x)/cos x ). Compute the derivative:( f'(x) = frac{-1 + 3 sin x}{cos^2 x} ). Since ( sin x > 1/2 ), ( 3 sin x > 3/2 ), so ( -1 + 3 sin x > 1/2 > 0 ). Hence, derivative is positive. Similarly, if ( a = 2 ), as above, derivative is positive. If ( a = 1.5 ), then ( a sin x ) when ( x ) is near ( pi/3 ), ( sin x approx sqrt{3}/2 approx 0.866 ), so ( 1.5 * 0.866 approx 1.299 ), so ( -1 + 1.299 approx 0.299 > 0 ). But when ( x ) is near ( pi/6 ), ( sin x approx 0.5 ), so ( 1.5 * 0.5 = 0.75 ), so ( -1 + 0.75 = -0.25 < 0 ). So the derivative would be negative near ( pi/6 ), which violates the monotonically increasing condition. Therefore, even though ( a = 1.5 ) satisfies the condition at ( x = pi/3 ), it doesn't satisfy it throughout the entire interval.Therefore, the critical point is the lower bound of ( a ), which must be 2. Therefore, the range of ( a ) is ( [2, infty) ).But wait, let me check once more with ( a = 2 ):( f'(x) = frac{-1 + 2 sin x}{cos^2 x} ). Since in ( (pi/6, pi/3) ), ( sin x in (1/2, sqrt{3}/2) ), so ( 2 sin x in (1, sqrt{3}) ), so ( -1 + 2 sin x in (0, sqrt{3} -1) ). Therefore, the numerator is always positive in the open interval, so the derivative is positive. Thus, ( a = 2 ) is acceptable, and any ( a > 2 ) would make the numerator even larger, hence derivative more positive. Therefore, the range is ( a geq 2 ).Wait, but the question says "the range of the real number ( a ) is ___". In Chinese math problems, sometimes intervals are written with square brackets or parentheses depending on inclusion. But here, since the original interval for ( x ) is open, but when ( a = 2 ), the derivative is positive everywhere in the open interval, so ( a = 2 ) is included. Therefore, the range should be ( [2, +infty) ).But let me check again if there is an upper limit. Suppose ( a ) is very large, say ( a = 1000 ). Then ( f'(x) = frac{-1 + 1000 sin x}{cos^2 x} ). Since ( sin x geq 1/2 ), so ( 1000 sin x geq 500 ), so ( -1 + 1000 sin x geq 499 ), which is positive. Hence, the derivative is positive. Therefore, there is no upper limit on ( a ); the larger ( a ), the more positive the derivative. Thus, the range of ( a ) is all real numbers greater than or equal to 2.Therefore, the final answer should be ( [2, +infty) ), which in boxed form is ( boxed{[2, +infty)} ).Final AnswerThe range of the real number ( a ) is boxed{[2, +infty)}.</think>To determine the range of the real number ( a ) such that the function ( f(x) = frac{a - sin x}{cos x} ) is monotonically increasing on the interval ( left( frac{pi}{6}, frac{pi}{3} right) ), we need to ensure that the derivative ( f'(x) ) is non-negative throughout this interval.First, we compute the derivative of ( f(x) ) using the quotient rule. Let ( u(x) = a - sin x ) and ( v(x) = cos x ). The derivatives are ( u'(x) = -cos x ) and ( v'(x) = -sin x ). Applying the quotient rule:[f'(x) = frac{(-cos x)(cos x) - (a - sin x)(-sin x)}{cos^2 x}]Simplifying the numerator:[- cos^2 x + (a sin x - sin^2 x) = -(cos^2 x + sin^2 x) + a sin x = -1 + a sin x]Thus, the derivative is:[f'(x) = frac{-1 + a sin x}{cos^2 x}]For ( f(x) ) to be monotonically increasing, ( f'(x) geq 0 ) on ( left( frac{pi}{6}, frac{pi}{3} right) ). Since ( cos^2 x > 0 ) in this interval, the numerator ( -1 + a sin x ) must be non-negative:[-1 + a sin x geq 0 implies a sin x geq 1 implies a geq frac{1}{sin x}]The function ( frac{1}{sin x} ) is decreasing on ( left( frac{pi}{6}, frac{pi}{3} right) ) because ( sin x ) is increasing. The maximum value of ( frac{1}{sin x} ) occurs at ( x = frac{pi}{6} ), which is ( 2 ). Therefore, ( a ) must be at least 2 to satisfy the inequality for all ( x ) in the interval.Hence, the range of the real number ( a ) is (boxed{[2, +infty)}).,<think>Okay, so I need to find the range of real number 'a' such that the function f(x) = (a - sin x)/cos x is monotonically increasing in the interval (π/6, π/3). Hmm, let's start by recalling that a function is monotonically increasing if its derivative is non-negative throughout the interval. So, my first step should be to find the derivative of f(x) and then ensure that it's greater than or equal to zero for all x in (π/6, π/3).Alright, let's compute the derivative f'(x). The function f(x) is a quotient of two functions: the numerator is (a - sin x) and the denominator is cos x. To find the derivative, I should use the quotient rule. The quotient rule states that if we have a function h(x) = g(x)/k(x), then h'(x) = [g'(x)k(x) - g(x)k'(x)] / [k(x)]². Applying this to f(x), let's set g(x) = a - sin x and k(x) = cos x. Then, the derivatives are:g'(x) = -cos x (since the derivative of a is zero and the derivative of -sin x is -cos x)k'(x) = -sin x (the derivative of cos x is -sin x)Plugging these into the quotient rule formula:f'(x) = [(-cos x)(cos x) - (a - sin x)(-sin x)] / (cos x)^2Let me simplify the numerator step by step:First term: (-cos x)(cos x) = -cos²xSecond term: (a - sin x)(-sin x) = -a sin x + sin²x. But there's a negative sign in front of this term in the quotient rule, so it becomes +a sin x - sin²x.Therefore, the numerator is -cos²x + a sin x - sin²x.Combine like terms. Since -cos²x - sin²x = - (cos²x + sin²x) = -1, because cos²x + sin²x = 1. So that simplifies the numerator to:-1 + a sin xSo, putting it all together:f'(x) = (-1 + a sin x) / cos²xTherefore, the derivative f'(x) is (a sin x - 1)/cos²x.Now, for the function to be monotonically increasing in (π/6, π/3), we need f'(x) ≥ 0 for all x in that interval. Since the denominator cos²x is always positive in the interval (π/6, π/3) because cos x is positive there (since both π/6 and π/3 are in the first quadrant where cosine is positive), the sign of f'(x) depends solely on the numerator: a sin x - 1 ≥ 0.Therefore, the condition simplifies to:a sin x - 1 ≥ 0 ⇒ a sin x ≥ 1 ⇒ a ≥ 1 / sin xBut this must hold for all x in (π/6, π/3). So, a must be greater than or equal to the maximum value of 1/sin x in that interval. Because if a is greater than or equal to the maximum of 1/sin x, then it will satisfy a ≥ 1/sin x for all x in the interval.Therefore, the range of a is [ max{1/sin x | x ∈ (π/6, π/3)} , ∞ )But now, I need to find the maximum value of 1/sin x over the interval (π/6, π/3). Since sin x is increasing in the interval (0, π/2), which includes (π/6, π/3), the function sin x is increasing here. Therefore, 1/sin x is decreasing on this interval because as sin x increases, 1/sin x decreases.Therefore, the maximum of 1/sin x occurs at the left endpoint of the interval, which is x = π/6.So, sin(π/6) = 1/2, so 1/sin(π/6) = 2. On the other hand, at x = π/3, sin(π/3) = √3/2 ≈ 0.866, so 1/sin(π/3) = 2/√3 ≈ 1.1547. Therefore, since 1/sin x is decreasing, the maximum value on the interval [π/6, π/3] is 2. However, the interval given is open (π/6, π/3), but since 1/sin x is continuous there, the supremum is still 2, but it's not attained in the open interval. However, since we need a ≥ 1/sin x for all x in the open interval, and 1/sin x approaches 2 as x approaches π/6 from the right, we need a to be at least 2 to satisfy the inequality for all x in (π/6, π/3). Similarly, as x approaches π/3 from the left, 1/sin x approaches 2/√3, but since 2 is larger, the critical point is a ≥ 2.Therefore, the range of real numbers a is [2, ∞). But wait, let me check.Wait, if the interval is open, does that affect anything? For example, even though x approaches π/6 from the right, sin x approaches 1/2, so 1/sin x approaches 2. So, for all x in (π/6, π/3), the maximum value that 1/sin x can get arbitrarily close to is 2, but never actually reaches 2. Therefore, if a is equal to 2, then a sin x -1 would be 2 sin x -1. At x approaching π/6 from the right, 2 sin x approaches 2*(1/2) =1, so 2 sin x -1 approaches 0. Therefore, at points near π/6, the derivative f'(x) approaches 0. But we need the derivative to be non-negative in the entire interval. So, as long as a sin x -1 ≥0 for all x in (π/6, π/3). If a =2, then 2 sin x -1 ≥0 ⇨ sin x ≥1/2. But in the interval (π/6, π/3), sin x is greater than or equal to 1/2 (since at π/6, sin x is 1/2, and increasing to √3/2 at π/3). So, in the open interval (π/6, π/3), sin x is greater than 1/2. Therefore, 2 sin x -1 would be greater than 0 in that interval. Wait, hold on. If x is in (π/6, π/3), then sin x is in (1/2, √3/2). Therefore, 2 sin x is in (1, √3), so 2 sin x -1 is in (0, √3 -1). Therefore, if a=2, then a sin x -1 = 2 sin x -1 >0 for all x in (π/6, π/3). Therefore, even at the left endpoint (though it's open), as x approaches π/6 from the right, the expression approaches 0, but is still positive inside the interval. Therefore, a=2 is acceptable. So, the minimum value of a is 2.Therefore, the range of a is [2, ∞). So, the answer should be a ≥ 2, which in boxed form is boxed{[2, +infty)}.But wait, let me verify this once again. Let's take a=2. Then f'(x) = (2 sin x -1)/cos²x. Since in (π/6, π/3), sin x >1/2, so 2 sin x -1 >0. Therefore, the derivative is positive throughout the interval. Hence, the function is indeed increasing when a=2. If a is slightly less than 2, say a=1.9, then near x=π/6, sin x approaches 1/2, so a sin x approaches 1.9*(1/2)=0.95, which is less than 1. Therefore, a sin x -1 would be negative near x=π/6, making the derivative negative there, which would contradict the function being monotonically increasing. Therefore, a must be at least 2. Hence, the range is [2, ∞). That seems correct.Final AnswerThe range of the real number ( a ) is boxed{[2, +infty)}.</think>To determine the range of the real number ( a ) such that the function ( f(x) = frac{a - sin x}{cos x} ) is monotonically increasing in the interval ( left( frac{pi}{6}, frac{pi}{3} right) ), we need to ensure that the derivative ( f'(x) ) is non-negative throughout this interval.First, we compute the derivative of ( f(x) ) using the quotient rule. Let ( g(x) = a - sin x ) and ( k(x) = cos x ). The derivatives are:[g'(x) = -cos x quad text{and} quad k'(x) = -sin x]Applying the quotient rule:[f'(x) = frac{g'(x)k(x) - g(x)k'(x)}{[k(x)]^2} = frac{(-cos x)(cos x) - (a - sin x)(-sin x)}{cos^2 x}]Simplifying the numerator:[-cos^2 x + (a sin x - sin^2 x) = -cos^2 x - sin^2 x + a sin x = -1 + a sin x]Thus, the derivative is:[f'(x) = frac{a sin x - 1}{cos^2 x}]For ( f(x) ) to be monotonically increasing, ( f'(x) geq 0 ). Since ( cos^2 x > 0 ) in the interval ( left( frac{pi}{6}, frac{pi}{3} right) ), the numerator ( a sin x - 1 ) must be non-negative:[a sin x - 1 geq 0 implies a geq frac{1}{sin x}]The function ( frac{1}{sin x} ) is decreasing in the interval ( left( frac{pi}{6}, frac{pi}{3} right) ) because ( sin x ) is increasing. Therefore, the maximum value of ( frac{1}{sin x} ) occurs at the left endpoint ( x = frac{pi}{6} ):[sin left( frac{pi}{6} right) = frac{1}{2} implies frac{1}{sin left( frac{pi}{6} right)} = 2]Thus, ( a ) must be at least 2 to satisfy the inequality for all ( x ) in ( left( frac{pi}{6}, frac{pi}{3} right) ).Therefore, the range of the real number ( a ) is (boxed{[2, +infty)}).]

🔑:To solve, we find the derivative of f(x): f'(x) = frac{-cos^2x - (a - sin x)(-sin x)}{cos^2x} = frac{-cos^2x + asin x - sin^2x}{cos^2x} = frac{asin x - 1}{cos^2x}.For f(x) to be monotonically increasing in the interval left(frac{pi}{6}, frac{pi}{3}right), f'(x) geq 0 must always hold, i.e., asin x - 1 geq 0 must always hold in the interval left(frac{pi}{6}, frac{pi}{3}right), which means asin x geq 1.Therefore, a geq frac{1}{sin x}.Since frac{pi}{6} < x < frac{pi}{3}, it follows that frac{1}{2} < sin x < frac{sqrt{3}}{2}, and thus frac{2sqrt{3}}{3} < frac{1}{sin x} < 2.Hence, a geq 2.Therefore, the answer is: boxed{[2, +infty)}.To solve this problem, we calculate the derivative of the function, establish an inequality based on the relationship between the monotonicity of the function and its derivative, and solve it using the method of separating variables. This problem mainly examines the relationship between the monotonicity of a function and its derivative, as well as the issue of the inequality always holding true. The key to solving this problem is the use of the method of separating variables, testing the students' computational and transformation skills.

🧠:[<think>Okay, let's see. The problem says that we have a sequence {a_n} defined by the recurrence relation a_{n+1} = 1/(2 - a_n). We need to find the range of initial values a_1 such that for any positive integer n, a_{n+1} > a_n. So, the sequence is strictly increasing. Hmm, interesting.First, maybe I should start by understanding how the sequence behaves. If the sequence is strictly increasing, then each term must be greater than the previous one. So, a_2 > a_1, a_3 > a_2, and so on. Let me write down the first few terms to get a sense.Given a_1, then a_2 = 1/(2 - a_1). Then a_3 = 1/(2 - a_2) = 1/(2 - 1/(2 - a_1)), and so on. This seems like a recursive sequence that could converge to a fixed point if it's convergent. But since it's strictly increasing, maybe it's converging to some limit? Wait, if it's increasing and bounded above, then it would converge. But the problem says it's increasing for any n, so maybe we need to ensure that it's bounded above as well? Otherwise, if it's increasing without bound, that might not be possible because of the recurrence relation.Let me think. Let's suppose that the sequence does converge to a limit L. Then, taking the limit on both sides of the recurrence relation, we would have L = 1/(2 - L). Solving this equation for L: multiplying both sides by (2 - L) gives L(2 - L) = 1 => 2L - L^2 = 1 => L^2 - 2L + 1 = 0 => (L - 1)^2 = 0 => L = 1. So, the fixed point is 1. That means if the sequence converges, it will converge to 1. But our sequence is strictly increasing. So, if it's approaching 1 from below, then all terms must be less than 1. Alternatively, if it's approaching 1 from above, but since the sequence is increasing, if it starts above 1, then it would have to increase towards infinity, but that might not be possible. Wait, let's check.Suppose a_1 is greater than 1. Then a_2 = 1/(2 - a_1). If a_1 > 1, then 2 - a_1 < 1, so the denominator is less than 1. But if a_1 is greater than 1, then 2 - a_1 is positive only if a_1 < 2. So, if a_1 is between 1 and 2, then 2 - a_1 is between 0 and 1, so a_2 = 1/(something between 0 and 1) would be greater than 1. So a_2 > 1. Then a_3 = 1/(2 - a_2). But since a_2 > 1, 2 - a_2 < 1, so a_3 = 1/(something less than 1) would again be greater than 1. Wait, so if a_1 is between 1 and 2, then all subsequent terms are greater than 1. But then, is the sequence increasing?Wait, let's take an example. Suppose a_1 = 1.5. Then a_2 = 1/(2 - 1.5) = 1/0.5 = 2. Then a_3 = 1/(2 - 2) = 1/0, which is undefined. Wait, so if a_1 is 1.5, a_2 is 2, then a_3 would be undefined. So that's a problem. Therefore, a_1 cannot be 1.5. So, if a_1 is between 1 and 2, then a_2 is greater than 1, but the denominator could approach zero, making the terms go to infinity. So, actually, if a_n approaches 2 from below, then a_{n+1} would approach 1/(2 - 2) which is infinity. But if the sequence is increasing, starting from a_1 in (1, 2), then a_2 = 1/(2 - a_1). Let's compute a_2 - a_1: 1/(2 - a_1) - a_1. If this is positive, then a_2 > a_1.So, maybe we need to find the values of a_1 such that each term is increasing, i.e., a_{n+1} > a_n for all n. So, starting with a_1, then a_2 > a_1, a_3 > a_2, etc.So, first, let's find the condition that a_2 > a_1. That is, 1/(2 - a_1) > a_1. Let's solve this inequality.1/(2 - a_1) > a_1.Multiply both sides by (2 - a_1). But we have to be careful about the sign of (2 - a_1). If (2 - a_1) is positive, then the inequality remains the same direction. If it's negative, the inequality flips.Case 1: 2 - a_1 > 0 => a_1 < 2.Then multiplying both sides by (2 - a_1) gives 1 > a_1(2 - a_1).So, 1 > 2a_1 - a_1^2.Rewriting: a_1^2 - 2a_1 + 1 > 0.Which factors as (a_1 - 1)^2 > 0. This is true for all a_1 ≠ 1. But since a_1 < 2, then the inequality 1/(2 - a_1) > a_1 holds if a_1 < 2 and a_1 ≠ 1. But wait, if a_1 < 2, but also, we need to check if the inequality is satisfied.Wait, but (a_1 - 1)^2 > 0 is equivalent to a_1 ≠ 1. So, in Case 1 where a_1 < 2, the inequality 1/(2 - a_1) > a_1 is equivalent to a_1 ≠ 1. But does that mean that when a_1 < 2, the inequality holds except when a_1 = 1?But let's test with a specific value. Suppose a_1 = 0. Then a_2 = 1/(2 - 0) = 0.5. Then a_2 > a_1? 0.5 > 0, yes. So, holds. If a_1 = 1, then a_2 = 1/(2 - 1) = 1. So, a_2 = a_1, which does not satisfy a_2 > a_1. So, the inequality is strict. So, a_1 =1 is excluded. If a_1 = 0.5, then a_2 = 1/(2 - 0.5) = 1/1.5 ≈ 0.666..., which is greater than 0.5. So, holds. If a_1 approaches 2 from below, say a_1 = 1.9, then a_2 = 1/(2 - 1.9) = 1/0.1 = 10. So, a_2 = 10 > 1.9. So, holds.But wait, when a_1 is between 1 and 2, we saw that a_2 = 1/(2 - a_1) which is greater than 1. But then, if the next term a_3 = 1/(2 - a_2), but a_2 is already greater than 1. If a_2 is greater than 1, then 2 - a_2 is less than 1. So, a_3 = 1/(2 - a_2) would be greater than 1/(2 - 1) = 1, but let's check.Wait, take a_1 = 1.5. Then a_2 = 2, as before. But then a_3 = 1/(2 - 2) is undefined. So, that's a problem. So, if a_1 is such that a_n reaches 2 for some n, the sequence terminates. Therefore, to have the sequence defined for all positive integers n, we need to ensure that a_n never equals 2 for any n. So, in particular, the denominator 2 - a_n must never be zero.Therefore, for the sequence to be defined for all n, we need that a_n < 2 for all n. So, we need to ensure that the sequence is bounded above by 2. Since the sequence is increasing, if it is bounded above by 2, it would converge to its supremum, which must be less than or equal to 2. But we know the fixed point is 1. So, if the sequence converges, it converges to 1. So, if the sequence is increasing and bounded above by 1, then it converges to 1. Wait, but if the sequence is increasing, and converges to 1, then all terms must be less than 1. Alternatively, if it starts above 1, but then it would have to increase, but since it can't exceed 2 (as that would make the next term undefined), but if it starts between 1 and 2, then as we saw with a_1 = 1.5, it might actually diverge to infinity if it reaches 2. But that's a problem.Wait, perhaps the sequence is increasing and approaches 1. So, maybe starting below 1. Let's test with a_1 = 0. Then the sequence is 0, 0.5, 1/(2 - 0.5) = 2/3 ≈ 0.666..., then 1/(2 - 2/3) = 1/(4/3) = 3/4 = 0.75, then 1/(2 - 3/4) = 1/(5/4) = 0.8, then 1/(2 - 0.8) = 1/1.2 ≈ 0.833..., and so on. Each term is increasing, approaching 1. So, in this case, starting from a_1 = 0, the sequence is increasing and converges to 1. So, that works.But if we take a_1 = 1.5, as before, the sequence goes to infinity at the third term. So, that's not allowed. So, if we start with a_1 in (1, 2), the sequence might not be defined for all n. So, maybe the starting value a_1 must be less than 1?Wait, let's check another value. Suppose a_1 = 0.5. Then a_2 = 1/(2 - 0.5) = 1/1.5 ≈ 0.666..., which is greater than 0.5. Then a_3 = 1/(2 - 0.666...) ≈ 1/1.333... ≈ 0.75, which is greater than 0.666..., and so on. Each term is increasing and approaching 1. So, that works.If a_1 is exactly 1, then a_2 = 1/(2 - 1) = 1, so the sequence is constant at 1. But the problem requires that a_{n+1} > a_n for all n, so a_1 = 1 is not allowed.If a_1 is less than 1, say a_1 = 0. Let's see, we already did that. So, starting from below 1, the sequence increases towards 1. If a_1 is between 1 and 2, the sequence might either diverge or not be strictly increasing?Wait, let's take a_1 = 1.2. Then a_2 = 1/(2 - 1.2) = 1/0.8 = 1.25. Then a_3 = 1/(2 - 1.25) = 1/0.75 ≈ 1.333..., then a_4 = 1/(2 - 1.333...) ≈ 1/0.666... ≈ 1.5, then a_5 = 1/(2 - 1.5) = 2, then a_6 is undefined. So, starting at 1.2, which is between 1 and 2, the sequence increases until it hits 2 at a_5, which is undefined. Therefore, the sequence isn't defined for all n. So, a_1 cannot be in (1, 2).If a_1 is greater than 2, let's see. Then 2 - a_1 is negative, so a_2 = 1/(negative) is negative. Then a_3 = 1/(2 - a_2). Since a_2 is negative, 2 - a_2 is greater than 2, so a_3 is positive but less than 1/2. Then a_4 = 1/(2 - a_3). Since a_3 is positive and less than 1/2, 2 - a_3 is greater than 1.5, so a_4 is less than 1/1.5 ≈ 0.666... Then a_4 is positive, less than 0.666..., so a_5 = 1/(2 - a_4) which is greater than 1/(2 - 0.666...) ≈ 0.75, which is greater than a_4. So, if a_1 is greater than 2, the sequence alternates between negative and positive terms? Wait, let's take a concrete example.Take a_1 = 3. Then a_2 = 1/(2 - 3) = -1. Then a_3 = 1/(2 - (-1)) = 1/3 ≈ 0.333..., a_4 = 1/(2 - 1/3) = 1/(5/3) = 3/5 = 0.6, a_5 = 1/(2 - 0.6) = 1/1.4 ≈ 0.714..., a_6 = 1/(2 - 0.714) ≈ 1/1.286 ≈ 0.778, and so on. So, starting from a_1 = 3, the sequence goes negative, then positive, and then seems to be increasing towards 1. But wait, in this case, after a_2, the terms become positive and increasing. But the problem states that a_{n+1} > a_n for all n. However, in this case, a_2 = -1 < a_1 = 3. So, the first step is a decrease, which violates the condition. Therefore, a_1 cannot be greater than 2.Similarly, if a_1 is negative, say a_1 = -1. Then a_2 = 1/(2 - (-1)) = 1/3 ≈ 0.333..., which is greater than a_1 (-1). Then a_3 = 1/(2 - 1/3) = 3/5 = 0.6 > 0.333..., a_4 = 1/(2 - 0.6) = 1/1.4 ≈ 0.714..., etc., increasing towards 1. So, starting from a negative a_1, the sequence becomes positive and increases towards 1. However, in this case, the first term a_1 is negative, a_2 is positive, so a_2 > a_1. Then a_3 > a_2, and so on. So, if a_1 is negative, the sequence is increasing from a_2 onwards, but is it increasing for all n? Let's check:a_1 = -1, a_2 = 1/3, which is greater than a_1. a_3 = 3/5 ≈ 0.6 > 1/3 ≈ 0.333..., a_4 ≈ 0.714 > 0.6, etc. So, seems okay. But wait, what if a_1 is very negative? Let's take a_1 = -10. Then a_2 = 1/(2 - (-10)) = 1/12 ≈ 0.083..., which is greater than a_1 (-10). Then a_3 = 1/(2 - 1/12) = 1/(23/12) = 12/23 ≈ 0.5217, which is greater than 0.083..., then a_4 = 1/(2 - 12/23) = 1/(34/23) = 23/34 ≈ 0.676, then a_5 = 1/(2 - 23/34) = 1/(45/34) = 34/45 ≈ 0.755..., and so on. So, even starting from a very negative a_1, the sequence becomes positive and increases towards 1. So, the question is, is the sequence {a_n} increasing for all n if a_1 is negative?But wait, the problem says "for any positive integer n, a_{n+1} > a_n". So, starting from a negative a_1, we have a_2 positive, which is greater than a_1 (since a_1 is negative). Then a_3 is greater than a_2, etc. So, in that case, the sequence is increasing for all n. So, perhaps a_1 can be any real number less than 1? Wait, but earlier when a_1 was between 1 and 2, the sequence became undefined after a few terms. So, we need to ensure that a_n < 2 for all n. So, if the sequence is increasing and converges to 1, which is less than 2, then if the starting term is less than 1, the sequence is bounded above by 1, right? Wait, no. If a_1 is less than 1, then the sequence increases towards 1, so all terms are less than 1. Therefore, 2 - a_n is always greater than 1, so a_{n+1} = 1/(2 - a_n) is between 1/2 and 1. Wait, but if a_1 is negative, a_2 is positive but less than 1. For example, a_1 = -1: a_2 = 1/3, which is positive and less than 1. Then a_3 = 1/(2 - 1/3) = 3/5 = 0.6, still less than 1. Then a_4 = 1/(2 - 0.6) = 1/1.4 ≈ 0.714, and so on, approaching 1. So, all terms are less than 1. Therefore, 2 - a_n is always greater than 1, so a_{n+1} is between 0.5 and 1. Therefore, the sequence is bounded above by 1 and increasing, so it converges to 1. Therefore, if a_1 is less than 1, the sequence is increasing and defined for all n. However, if a_1 is between 1 and 2, the sequence is increasing but may hit 2 at some term, making the next term undefined. So, to ensure that the sequence is defined for all n and strictly increasing, a_1 must be less than 1. Wait, but when a_1 is less than 1, the sequence is increasing and converges to 1. So, in that case, all terms are less than 1, so 2 - a_n is greater than 1, hence a_{n+1} is between 1/1 and 1/(2 - a_n). Wait, maybe not. Wait, if a_n < 1, then 2 - a_n > 1, so 1/(2 - a_n) < 1. Wait, but if a_n is approaching 1 from below, then 1/(2 - a_n) approaches 1/(2 - 1) = 1. So, as a_n approaches 1, a_{n+1} approaches 1. So, the sequence converges to 1, and all terms are less than 1. Therefore, the sequence is bounded above by 1 and increasing. Therefore, if a_1 < 1, then {a_n} is strictly increasing and converges to 1. If a_1 is exactly 1, the sequence is constant at 1. If a_1 > 1, then as we saw, the sequence may become undefined or not increasing for all n.But wait, in the case where a_1 is negative, say a_1 = -10, the first term is -10, the second term is 1/12 ≈ 0.083, which is greater than -10, so a_2 > a_1. Then each subsequent term is increasing. So, even if a_1 is negative, as long as a_1 < 1, the sequence is strictly increasing. Therefore, the range of a_1 is all real numbers less than 1. But wait, let's check if there are any other constraints.Wait, suppose a_1 is less than 1, but in the case where the sequence could potentially decrease. Wait, no. If a_1 < 1, then 1/(2 - a_1) > a_1. Because when we solved the inequality 1/(2 - a_1) > a_1 for a_1 < 2, we found that it's true as long as a_1 ≠ 1. But since a_1 < 1, then it's certainly true. So, a_2 > a_1. Then, since a_2 = 1/(2 - a_1), and since a_1 < 1, then 2 - a_1 > 1, so a_2 = 1/(2 - a_1) < 1. Therefore, a_2 is still less than 1. So, the same logic applies: a_3 = 1/(2 - a_2) > a_2 because a_2 < 1. So, by induction, if a_n < 1, then a_{n+1} = 1/(2 - a_n) > a_n and a_{n+1} < 1. Therefore, by induction, the sequence is strictly increasing and bounded above by 1, hence converges to 1. Therefore, all a_1 < 1 will result in a strictly increasing sequence converging to 1.But wait, what about a_1 = 0.5? Then the sequence is 0.5, 1/(2 - 0.5)= 0.666..., 1/(2 - 0.666...)≈0.75, 1/(2 - 0.75)=0.8, etc., each term increasing towards 1. So, that works.What if a_1 is something like 1.5? Wait, 1.5 is greater than 1, but we already saw that starting from 1.5, the sequence goes 2, then undefined. So, it's invalid. Therefore, a_1 must be less than 1. But wait, even if a_1 is less than 1 but greater than some other number? Let's see. Suppose a_1 is 0.9. Then a_2 = 1/(2 - 0.9) = 1/1.1 ≈ 0.909..., which is greater than 0.9. Then a_3 = 1/(2 - 0.909...) ≈ 1/1.0909... ≈ 0.916..., and so on, approaching 1. So, works. If a_1 is 0.99, then a_2 ≈ 1/(2 - 0.99) ≈ 1/1.01 ≈ 0.990099..., which is greater than 0.99. Then a_3 ≈ 1/(2 - 0.990099) ≈ 1/1.0099 ≈ 0.990196..., so still increasing. So, even very close to 1, starting from below, the sequence increases towards 1.Therefore, it seems that as long as a_1 is less than 1, the sequence is strictly increasing and remains less than 1. Hence, the range of a_1 is (-∞, 1). But wait, the problem says "the range of values for a_1". But in the problem statement, there's no restriction on a_1 being real or anything else. But since the problem is given in a mathematical context, we can assume a_1 is a real number.But wait, let's check if there's a lower bound. For example, can a_1 be any real number less than 1? Let's take a_1 approaching negative infinity. Then a_2 = 1/(2 - a_1). If a_1 approaches negative infinity, then 2 - a_1 approaches positive infinity, so a_2 approaches 0 from the positive side. So, for very large negative a_1, a_2 is a small positive number. Then a_3 = 1/(2 - a_2) ≈ 1/2, which is greater than a_2. Then a_4 ≈ 1/(2 - 1/2) = 2/3, which is greater than 1/2, and so on. So, even if a_1 is a large negative number, the sequence still starts increasing once it becomes positive. Therefore, even if a_1 is very negative, the sequence is increasing for all n. Therefore, the lower bound is negative infinity. Therefore, the range of a_1 is all real numbers less than 1.But wait, let's check the case when a_1 is exactly 1. If a_1 =1, then a_2 = 1/(2 -1)=1, so the sequence is constant. But the problem requires a_{n+1} > a_n for all n, so a_1 cannot be 1. So, the range is a_1 <1.But let's confirm once more. Suppose a_1 is less than 1. Then:1. The first term a_1 <1.2. The second term a_2 =1/(2 - a_1). Since a_1 <1, 2 -a_1 >1, so a_2 <1. Also, since a_1 <1, then 1/(2 -a_1) > a_1. Because solving 1/(2 -a_1) >a_1 gives (a_1 -1)^2 >0, which is true when a_1 ≠1. But since a_1 <1, the inequality holds, so a_2 >a_1.3. Then, since a_2 <1 and a_2 >a_1, the same applies to a_3: a_3 =1/(2 -a_2) >a_2 and a_3 <1.By induction, this continues: each term a_n <1, so a_{n+1} =1/(2 -a_n) >a_n and a_{n+1} <1. Therefore, the entire sequence is strictly increasing and bounded above by1, hence converges to1. Therefore, the initial term a_1 can be any real number less than1.Therefore, the range of values for a_1 is all real numbers less than1, which in interval notation is (-∞, 1).But wait, the problem is in Chinese, so maybe the answer is supposed to be in a different format? But the user just said "put your final answer within boxed{}". So, I need to write the interval. So, the answer is (-∞, 1). Therefore, boxed, it's boxed{(-infty, 1)}.Wait, but let me check once more. Suppose a_1 is 0. Let's compute a few terms:a_1 = 0a_2 = 1/(2 - 0) = 0.5 >0a_3 =1/(2 -0.5)=1/1.5≈0.666... >0.5a_4=1/(2 -0.666...)=1/1.333...≈0.75>0.666...a_5=1/(2 -0.75)=1/1.25=0.8>0.75Continues approaching1.Another example: a_1 =-1a_2 =1/(2 -(-1))=1/3≈0.333...>-1a_3=1/(2 -1/3)=1/(5/3)=3/5=0.6>0.333...a_4=1/(2 -0.6)=1/1.4≈0.714>0.6a_5≈1/(2 -0.714)=1/1.286≈0.778>0.714Approaching1.So, works. If a_1 =1, the sequence is constant at1, which is not strictly increasing. If a_1 >1, the sequence either becomes undefined or isn't increasing for all n. So, conclusion: a_1 must be less than1. Therefore, the answer is (-∞,1).Final AnswerThe range of values for (a_{1}) is boxed{(-infty, 1)}.</think>Given the sequence ({a_n}) defined by the recurrence relation (a_{n+1} = frac{1}{2 - a_n}), we need to find the range of initial values (a_1) such that (a_{n+1} > a_n) for all positive integers (n).1. Fixed Point Analysis: - If the sequence converges to a limit (L), then (L = frac{1}{2 - L}). - Solving this equation gives (L = 1). Therefore, the sequence converges to 1 if it is convergent.2. Behavior of the Sequence: - For the sequence to be strictly increasing, each term (a_{n+1}) must be greater than (a_n). - We start by solving the inequality (a_2 > a_1): [ frac{1}{2 - a_1} > a_1 ] - Multiplying both sides by (2 - a_1) (assuming (2 - a_1 > 0) i.e., (a_1 < 2)) gives: [ 1 > a_1(2 - a_1) implies a_1^2 - 2a_1 + 1 > 0 implies (a_1 - 1)^2 > 0 ] - This inequality holds for all (a_1 neq 1). Since we need (a_1 < 2), the inequality holds for (a_1 < 1).3. Checking the Interval: - If (a_1 < 1), then (2 - a_1 > 1), ensuring (a_2 = frac{1}{2 - a_1}) is positive and less than 1. - By induction, if (a_n < 1), then (a_{n+1} = frac{1}{2 - a_n}) is greater than (a_n) and less than 1. Thus, the sequence is strictly increasing and bounded above by 1.4. Negative Initial Values: - If (a_1) is negative, subsequent terms become positive and the sequence remains increasing. For example, (a_1 = -1) leads to (a_2 = frac{1}{3}), which is positive and increasing.5. Conclusion: - The sequence is strictly increasing and defined for all (n) if and only if (a_1 < 1). If (a_1 geq 1), the sequence either becomes undefined or is not strictly increasing.Thus, the range of values for (a_1) is (boxed{(-infty, 1)}).,<think>Okay, so I need to figure out the range of values for the first term ( a_1 ) in the sequence defined by ( a_{n+1} = frac{1}{2 - a_n} ), such that the sequence is strictly increasing for all positive integers ( n ). Hmm, let's start by understanding the problem step by step.First, the sequence is defined recursively, meaning each term depends on the previous one. The condition given is that every subsequent term must be greater than the previous one, so ( a_{n+1} > a_n ) for all ( n geq 1 ). My goal is to find all possible starting values ( a_1 ) that satisfy this condition indefinitely.Let me start by writing down the first few terms in terms of ( a_1 ) to see if I can spot a pattern or find constraints on ( a_1 ).Given:( a_{n+1} = frac{1}{2 - a_n} )So,( a_2 = frac{1}{2 - a_1} )( a_3 = frac{1}{2 - a_2} = frac{1}{2 - frac{1}{2 - a_1}} )Let me compute ( a_3 ) step by step. Let's first compute the denominator:( 2 - a_2 = 2 - frac{1}{2 - a_1} = frac{(2)(2 - a_1) - 1}{2 - a_1} = frac{4 - 2a_1 - 1}{2 - a_1} = frac{3 - 2a_1}{2 - a_1} )Therefore,( a_3 = frac{1}{frac{3 - 2a_1}{2 - a_1}} = frac{2 - a_1}{3 - 2a_1} )Similarly, ( a_4 = frac{1}{2 - a_3} = frac{1}{2 - frac{2 - a_1}{3 - 2a_1}} )Again, let's compute the denominator:( 2 - frac{2 - a_1}{3 - 2a_1} = frac{2(3 - 2a_1) - (2 - a_1)}{3 - 2a_1} = frac{6 - 4a_1 - 2 + a_1}{3 - 2a_1} = frac{4 - 3a_1}{3 - 2a_1} )Therefore,( a_4 = frac{3 - 2a_1}{4 - 3a_1} )Hmm, so the terms seem to follow a pattern. Let's note them down:( a_1 = a_1 )( a_2 = frac{1}{2 - a_1} )( a_3 = frac{2 - a_1}{3 - 2a_1} )( a_4 = frac{3 - 2a_1}{4 - 3a_1} )Looking at the pattern, it seems like each subsequent term has a numerator and denominator that increment by 1 and the coefficient of ( a_1 ) increments by 1 as well. So perhaps:( a_n = frac{(n - 1) - (n - 2)a_1}{n - (n - 1)a_1} )Wait, let me check if this holds for the first few terms.For ( n = 1 ), according to the formula:( a_1 = frac{0 - (-1)a_1}{1 - 0a_1} = frac{a_1}{1} = a_1 ). That's correct.For ( n = 2 ):( a_2 = frac{1 - 0a_1}{2 - 1a_1} = frac{1}{2 - a_1} ). Correct.For ( n = 3 ):( a_3 = frac{2 - 1a_1}{3 - 2a_1} ). Which matches our earlier calculation.Similarly, ( n = 4 ):( a_4 = frac{3 - 2a_1}{4 - 3a_1} ). Also correct.So, it seems like the general term is:( a_n = frac{(n - 1) - (n - 2)a_1}{n - (n - 1)a_1} )Therefore, if I can express ( a_n ) in terms of ( a_1 ), I can analyze the inequalities ( a_{n+1} > a_n ) for all ( n geq 1 ). However, verifying this formula for all ( n ) might be complicated. Alternatively, maybe I can use induction to confirm that this formula is valid. But perhaps there's a better approach.Alternatively, maybe we can analyze the recurrence relation itself. Since the sequence is defined by ( a_{n+1} = frac{1}{2 - a_n} ), and we need the sequence to be strictly increasing. So, for the sequence to be increasing, each term must be greater than the previous one. So, ( a_{n+1} > a_n ) for all ( n ).Therefore, the inequality ( frac{1}{2 - a_n} > a_n ) must hold for all ( n geq 1 ). Let's write that inequality down:( frac{1}{2 - a_n} > a_n )To solve this inequality, we can consider two cases depending on the denominator ( 2 - a_n ). The denominator can be positive or negative, but since we have a fraction, we need to be careful about the sign.Case 1: ( 2 - a_n > 0 ) (i.e., ( a_n < 2 ))In this case, we can multiply both sides of the inequality by ( 2 - a_n ) without changing the inequality direction:( 1 > a_n(2 - a_n) )( 1 > 2a_n - a_n^2 )( a_n^2 - 2a_n + 1 > 0 )( (a_n - 1)^2 > 0 )Which is true as long as ( a_n neq 1 ).Case 2: ( 2 - a_n < 0 ) (i.e., ( a_n > 2 ))In this case, multiplying both sides by ( 2 - a_n ) (which is negative) reverses the inequality:( 1 < a_n(2 - a_n) )( 1 < 2a_n - a_n^2 )( a_n^2 - 2a_n + 1 < 0 )( (a_n - 1)^2 < 0 )But a square is always non-negative, so this inequality can never be true. Therefore, Case 2 is impossible.Therefore, for the inequality ( a_{n+1} > a_n ) to hold, it must be that ( a_n < 2 ) and ( a_n neq 1 ). However, since we need this to hold for all ( n geq 1 ), we need that all terms ( a_n < 2 ) and ( a_n neq 1 ).But wait, the problem states "for any positive integer ( n ), ( a_{n+1} > a_n )", so we need to ensure that all terms satisfy ( a_{n} < 2 ), and ( a_n neq 1 ). But also, since the sequence is increasing, we need to make sure that the terms do not exceed 2 as ( n ) increases.But maybe there's a fixed point for this recurrence relation. Fixed points are values ( x ) such that ( x = frac{1}{2 - x} ). Solving for ( x ):( x(2 - x) = 1 )( 2x - x^2 = 1 )( x^2 - 2x + 1 = 0 )( (x - 1)^2 = 0 )Therefore, the only fixed point is ( x = 1 ). So if the sequence ever reaches 1, it will stay there. However, the problem states that the sequence is strictly increasing for all ( n ), so if it ever reaches 1, the subsequent terms would be 1, which contradicts the strictly increasing condition. Therefore, the sequence cannot reach 1. Therefore, ( a_n neq 1 ) for all ( n ).So, putting this together, each term must be less than 2 and not equal to 1, and the sequence is increasing. Let's think about possible starting values ( a_1 ).Since the sequence is increasing, ( a_1 < a_2 < a_3 < dots ), and all terms are less than 2. So, starting from ( a_1 ), each subsequent term is larger, approaching some limit. But since the fixed point is 1, which is the only fixed point, maybe the sequence converges to 1? Let me check.Suppose the sequence converges to some limit ( L ). Then, taking the limit on both sides of the recurrence:( L = frac{1}{2 - L} )Which gives ( L(2 - L) = 1 ), same as before, leading to ( L = 1 ). So if the sequence converges, it converges to 1. However, if the sequence is strictly increasing and all terms are less than 2, then if it converges to 1, the terms must be approaching 1 from below. But 1 is the fixed point. Alternatively, if the terms are approaching 1 from above, but they can't exceed 2, but given the recursion formula, if a term is greater than 1, how does the sequence behave?Wait, let's suppose ( a_n < 1 ). Then ( 2 - a_n > 1 ), so ( a_{n+1} = frac{1}{2 - a_n} < 1 ). So if ( a_n < 1 ), then ( a_{n+1} < 1 ). Similarly, if ( a_n > 1 ), then ( 2 - a_n < 1 ), so ( a_{n+1} = frac{1}{2 - a_n} > 1 ). So once a term is greater than 1, all subsequent terms are greater than 1. But since our sequence is supposed to be increasing, if ( a_1 < 1 ), then all subsequent terms are less than 1, and the sequence would be increasing towards 1. However, if ( a_1 > 1 ), then all subsequent terms are greater than 1, but since we need the sequence to be increasing and bounded above by 2, perhaps starting from a value between 1 and 2, but we need to check.Wait, but if ( a_1 > 1 ), then ( a_2 = frac{1}{2 - a_1} ). However, if ( a_1 ) is between 1 and 2, then ( 2 - a_1 ) is between 0 and 1, so ( a_2 = frac{1}{2 - a_1} > 1 ). So ( a_2 > 1 ). But then, is ( a_2 > a_1 )?Let's suppose ( a_1 ) is between 1 and 2. Let's pick a value, say ( a_1 = 1.5 ).Then ( a_2 = 1 / (2 - 1.5) = 1 / 0.5 = 2 ). Wait, but 2 is the upper bound. But the recurrence formula at ( a_n = 2 ) would give ( a_{n+1} = 1/(2 - 2) ), which is undefined. So ( a_1 = 1.5 ) gives ( a_2 = 2 ), which is not allowed. Therefore, if ( a_1 ) approaches 2 from below, ( a_2 ) approaches infinity? Wait, no. Wait, if ( a_n ) approaches 2 from below, then ( 2 - a_n ) approaches 0 from above, so ( a_{n+1} ) approaches positive infinity. Therefore, if ( a_n ) is close to 2, ( a_{n+1} ) becomes very large. But since our sequence is supposed to be increasing and all terms must be less than 2, this suggests that ( a_n ) cannot approach 2, which seems contradictory. Wait, perhaps my earlier conclusion that all terms must be less than 2 is incorrect.Wait, let's re-examine the inequality.We started with the condition that ( a_{n+1} > a_n ). To satisfy this, we considered the inequality ( frac{1}{2 - a_n} > a_n ), leading to ( (a_n - 1)^2 > 0 ), so ( a_n neq 1 ), and in the case where ( a_n < 2 ). However, if ( a_n geq 2 ), then ( 2 - a_n leq 0 ), which would make ( a_{n+1} ) undefined or negative. Since the problem states that ( a_{n+1} ) is defined for all positive integers ( n ), we must have that ( 2 - a_n neq 0 ), so ( a_n neq 2 ). Also, if ( a_n > 2 ), then ( 2 - a_n < 0 ), so ( a_{n+1} ) would be negative. However, the problem doesn't specify that the terms must be positive, but if ( a_{n+1} ) is negative, let's see if the sequence can still be increasing.Suppose ( a_n > 2 ), leading to ( a_{n+1} = frac{1}{2 - a_n} ), which is negative. Then, ( a_{n+1} ) is negative, so ( a_{n+1} < 0 < a_n ), but this would mean ( a_{n+1} < a_n ), contradicting the requirement that the sequence is increasing. Therefore, such a term ( a_n > 2 ) would cause the next term to drop below zero and violate the increasing condition. Therefore, to maintain the sequence increasing, we must have ( a_n < 2 ) for all ( n ).Therefore, all terms must lie in ( (-infty, 2) ), but since the sequence is increasing and ( a_{n+1} = frac{1}{2 - a_n} ), let's see the implications.But also, we need to ensure that each ( a_{n} ) is such that ( 2 - a_n neq 0 ), so ( a_n neq 2 ), which we already have from the previous reasoning.Furthermore, if ( a_n < 1 ), then ( a_{n+1} = frac{1}{2 - a_n} ). Since ( a_n < 1 ), ( 2 - a_n > 1 ), so ( a_{n+1} < 1 ). Therefore, if ( a_1 < 1 ), then all terms ( a_n < 1 ), and the sequence is bounded above by 1. However, since the sequence is increasing and bounded above by 1, it must converge to a limit ( L leq 1 ). But the only fixed point is 1, so it would converge to 1. However, convergence to 1 would imply that the terms approach 1, getting closer and closer. But if the sequence is strictly increasing and approaching 1, then all terms must be less than 1. Therefore, starting from ( a_1 < 1 ), the sequence is increasing and approaching 1. In this case, the condition ( a_{n+1} > a_n ) is satisfied because each term is getting closer to 1 from below. Therefore, such a starting value ( a_1 < 1 ) would satisfy the condition.Alternatively, if ( a_1 ) is between 1 and 2, then ( a_2 = frac{1}{2 - a_1} ). Let's take an example. Suppose ( a_1 = 1.5 ), then ( a_2 = 1 / (0.5) = 2 ), which is not allowed since ( a_2 ) would be 2, but ( a_{n} ) must be less than 2. Wait, but if ( a_1 ) approaches 1 from above, say ( a_1 = 1.1 ), then ( a_2 = 1 / (2 - 1.1) = 1 / 0.9 ≈ 1.111 ). Then, ( a_2 ≈ 1.111 > a_1 = 1.1 ). Then, ( a_3 = 1 / (2 - 1.111) ≈ 1 / 0.889 ≈ 1.125 ), which is greater than ( a_2 ). Continuing this, each term is increasing, approaching 2. But wait, if the terms are approaching 2, then as ( a_n ) approaches 2, ( a_{n+1} = 1 / (2 - a_n) ) approaches infinity. Therefore, if we start with ( a_1 ) in (1, 2), the sequence would increase towards infinity, which contradicts the requirement that all terms are less than 2. Therefore, such a starting value would cause the sequence to exceed 2 in finite steps, which is not allowed. Therefore, starting values in (1, 2) are invalid because they lead to terms exceeding 2, which would make subsequent terms undefined or negative, violating the increasing condition.Wait, but if ( a_1 ) is slightly above 1, say 1.1, then ( a_2 ≈ 1.111 ), which is still less than 2. Then ( a_3 ≈ 1.125 ), still less than 2. Wait, but as ( a_n ) approaches 2, ( a_{n+1} ) approaches infinity. So if the sequence is increasing and approaching 2, then ( a_{n} ) would actually approach 2, but then ( a_{n+1} ) would go to infinity. However, in reality, once ( a_n ) gets close enough to 2, the next term ( a_{n+1} ) becomes very large, but positive. However, if ( a_{n+1} ) is positive and very large, say greater than 2, then ( a_{n+2} = 1/(2 - a_{n+1}) ) would be negative, since ( 2 - a_{n+1} ) is negative. Therefore, the term after a large ( a_{n+1} ) would be negative, causing the sequence to decrease, which violates the strictly increasing condition. Therefore, starting from ( a_1 ) in (1, 2), the sequence would eventually reach a term greater than 2, leading to a negative term, which is a decrease. Therefore, such starting values are invalid.Therefore, the only valid starting values are those where ( a_1 < 1 ), leading to all terms ( a_n < 1 ), and the sequence is increasing towards 1, never exceeding 1. However, we need to check if there are any other constraints on ( a_1 ).For example, suppose ( a_1 ) is very small, say ( a_1 = 0 ). Then ( a_2 = 1/(2 - 0) = 1/2 ), which is greater than 0. Then ( a_3 = 1/(2 - 1/2) = 1/(1.5) ≈ 0.6667 ), which is greater than 1/2. Then ( a_4 = 1/(2 - 0.6667) ≈ 1/1.3333 ≈ 0.75 ), which is greater than 0.6667. Continuing this, the sequence approaches 1. So starting from ( a_1 = 0 ), the sequence is increasing and approaches 1. Therefore, ( a_1 = 0 ) is valid.But what if ( a_1 ) is negative? Let's test ( a_1 = -1 ). Then ( a_2 = 1/(2 - (-1)) = 1/3 ≈ 0.333 ). Then ( a_3 = 1/(2 - 1/3) = 1/(5/3) = 3/5 = 0.6 ). Then ( a_4 = 1/(2 - 0.6) = 1/1.4 ≈ 0.714 ), and so on. The sequence is increasing towards 1. So even starting from a negative ( a_1 ), as long as ( a_1 < 1 ), the sequence is increasing and approaches 1.But wait, if ( a_1 ) is negative, then ( a_2 = 1/(2 - a_1) ). Since ( a_1 < 1 ), and if ( a_1 ) is negative, ( 2 - a_1 > 2 ), so ( a_2 = 1/(2 - a_1) < 1/2 ). So for example, if ( a_1 = -2 ), then ( a_2 = 1/(2 - (-2)) = 1/4 = 0.25 ), then ( a_3 = 1/(2 - 0.25) = 1/1.75 ≈ 0.571 ), then ( a_4 ≈ 1/(2 - 0.571) ≈ 1/1.429 ≈ 0.7 ), and so on. So the sequence is increasing towards 1. Therefore, even negative starting values ( a_1 ) result in an increasing sequence approaching 1.However, we need to ensure that all terms satisfy ( a_{n+1} > a_n ). Let's check with a more negative ( a_1 ), say ( a_1 = -10 ).( a_2 = 1/(2 - (-10)) = 1/12 ≈ 0.0833 ). Then ( a_3 = 1/(2 - 1/12) = 1/(23/12) = 12/23 ≈ 0.5217 ), which is greater than ( a_2 ≈ 0.0833 ). Then ( a_4 = 1/(2 - 12/23) = 1/(34/23) = 23/34 ≈ 0.6765 ), which is greater than ( a_3 ≈ 0.5217 ). Then ( a_5 = 1/(2 - 23/34) = 1/(45/34) = 34/45 ≈ 0.7556 ), and so on. So the sequence is increasing each time, approaching 1. Therefore, even with a very negative ( a_1 ), the sequence increases towards 1.Therefore, perhaps the starting value ( a_1 ) can be any real number less than 1? But we need to check if there are any constraints that would prevent ( a_1 ) from being too low.Wait, but the recursion formula ( a_{n+1} = frac{1}{2 - a_n} ) requires that ( 2 - a_n neq 0 ), so ( a_n neq 2 ). Since we have already established that for the sequence to be increasing, all terms must be less than 2. However, if ( a_1 ) is less than 1, then as shown in the examples, the sequence increases towards 1, remaining below 1. Wait, no. Wait, in the previous examples, starting from ( a_1 = 0 ), the sequence goes 0, 0.5, 0.666..., 0.75, etc., approaching 1. Similarly, starting from ( a_1 = -1 ), the sequence approaches 1. So all terms are less than 1, but approaching 1. Therefore, starting from ( a_1 < 1 ), the sequence remains below 1 and increases towards 1.Wait, but in the case of ( a_1 < 1 ), all terms ( a_n < 1 ), and the sequence is increasing. Therefore, the condition ( a_{n+1} > a_n ) is satisfied as long as ( a_n neq 1 ). However, since ( a_n ) approaches 1, the terms get arbitrarily close to 1 but never reach it. Therefore, the sequence is well-defined, and all terms are less than 1. Therefore, for ( a_1 < 1 ), the sequence is strictly increasing and bounded above by 1, hence converges to 1.But the problem states "for any positive integer ( n ), ( a_{n+1} > a_n )", so the starting value ( a_1 ) must be such that this inequality holds for all ( n geq 1 ). So, to ensure that ( a_{n} < 1 ) for all ( n ), given that the sequence is increasing and approaching 1, we need to make sure that the starting value ( a_1 ) is such that the sequence does not reach or exceed 1 in finite steps. Since the sequence approaches 1 asymptotically, starting from ( a_1 < 1 ), the terms will never actually reach 1. Therefore, all ( a_1 < 1 ) would satisfy the condition ( a_{n+1} > a_n ) for all ( n ).But wait, let's check for the case when ( a_1 = 1 ). Then ( a_2 = 1/(2 - 1) = 1 ), so the sequence becomes constant at 1. But since the problem requires the sequence to be strictly increasing, ( a_1 = 1 ) is invalid. Therefore, ( a_1 ) cannot be 1.Now, what if ( a_1 = 1 )? As mentioned, the sequence becomes constant, which violates the strictly increasing condition. So ( a_1 neq 1 ).But what about values of ( a_1 ) approaching 1 from below? For example, ( a_1 = 0.999 ). Then ( a_2 = 1/(2 - 0.999) ≈ 1/1.001 ≈ 0.999001 ), which is slightly greater than ( a_1 ). Then ( a_3 ≈ 1/(2 - 0.999001) ≈ 1/1.000999 ≈ 0.999001999 ), which is slightly greater than ( a_2 ). Each term is getting closer to 1, but still less than 1. Therefore, even as ( a_1 ) approaches 1 from below, the sequence remains strictly increasing and all terms are less than 1. Therefore, ( a_1 ) can be any real number less than 1.But wait, let's check a starting value ( a_1 ) such that ( a_1 < 1 ), but maybe with some other constraints. For example, suppose ( a_1 = 1.5 ), but we saw earlier that this leads to a term equal to 2, which is invalid. But 1.5 is greater than 1, so it's already in the other case.But if ( a_1 < 1 ), then all subsequent terms ( a_n < 1 ), and the sequence is strictly increasing. Therefore, the range of ( a_1 ) is all real numbers less than 1. However, we need to ensure that the recursion doesn't lead to any undefined terms. Since ( a_1 < 1 ), then ( 2 - a_1 > 1 ), so ( a_2 = 1/(2 - a_1) ) is positive and less than 1. Then, ( a_3 = 1/(2 - a_2) ), since ( a_2 < 1 ), ( 2 - a_2 > 1 ), so ( a_3 < 1 ), and so on. Therefore, all terms are positive if ( a_1 ) is such that ( a_n > 0 ). Wait, but if ( a_1 ) is negative, ( a_2 = 1/(2 - a_1) ) is positive because ( 2 - a_1 > 2 - 0 = 2 ), so ( a_2 ) is positive. Then ( a_3 = 1/(2 - a_2) ). Since ( a_2 ) is positive and less than 1 (since ( a_1 < 1 )), then ( 2 - a_2 > 1 ), so ( a_3 = 1/(2 - a_2) < 1 ), but positive. Therefore, once ( a_1 ) is negative, ( a_2 ) becomes positive, and from there, the sequence continues to be positive and increasing towards 1.Therefore, even if ( a_1 ) is negative, the subsequent terms become positive and the sequence continues to increase towards 1. Therefore, as long as ( a_1 < 1 ), the sequence is strictly increasing and all terms are less than 1 (after the first term if ( a_1 ) is negative). Therefore, the starting value ( a_1 ) can be any real number less than 1.Wait, but let's verify with a starting value ( a_1 ) very negative. For example, ( a_1 = -1000 ).Then, ( a_2 = 1/(2 - (-1000)) = 1/1002 ≈ 0.000998 ), which is greater than ( a_1 = -1000 ). Then ( a_3 = 1/(2 - 0.000998) ≈ 1/1.999002 ≈ 0.50025 ), which is greater than ( a_2 ≈ 0.000998 ). Then ( a_4 = 1/(2 - 0.50025) ≈ 1/1.49975 ≈ 0.6669 ), greater than ( a_3 ≈ 0.50025 ). Then ( a_5 ≈ 1/(2 - 0.6669) ≈ 1/1.3331 ≈ 0.7501 ), and so on. Each term is increasing, approaching 1. So even with a very negative ( a_1 ), the sequence increases towards 1. Therefore, it seems like any ( a_1 < 1 ) works.But wait, what if ( a_1 = 2 )? Then ( a_2 ) is undefined. Similarly, if ( a_1 = 1 ), the sequence becomes constant. If ( a_1 > 1 ), as discussed earlier, the sequence would eventually exceed 2, leading to undefined terms or negative terms which would violate the increasing condition. Therefore, the only valid starting values are ( a_1 < 1 ).However, let's check if there's a lower bound. For example, can ( a_1 ) be any real number less than 1, or is there a lower limit?Suppose ( a_1 ) approaches negative infinity. Then ( a_2 = 1/(2 - a_1) ) approaches 0 from the positive side. Then ( a_3 = 1/(2 - a_2) ≈ 1/2 ), ( a_4 ≈ 1/(2 - 1/2) = 2/3 ), and so on, still increasing towards 1. Therefore, even as ( a_1 ) approaches negative infinity, the subsequent terms approach 0 and then increase towards 1. Therefore, there's no lower bound on ( a_1 ); it can be any real number less than 1.Therefore, the range of values for ( a_1 ) is all real numbers less than 1, i.e., ( (-infty, 1) ).But wait, the problem is presented in a mathematical context, possibly expecting a different answer. Let me double-check.Wait, let me check with another approach. Suppose we model the sequence as a function. Let ( f(x) = frac{1}{2 - x} ). Then the sequence is defined by ( a_{n+1} = f(a_n) ). We need ( f(x) > x ) for all ( x = a_n ).So, solving ( f(x) > x ):( frac{1}{2 - x} > x )Which we already did earlier, leading to ( (x - 1)^2 > 0 ), i.e., ( x neq 1 ), and ( x < 2 ).Therefore, for ( f(x) > x ), we need ( x < 2 ) and ( x neq 1 ).But since our sequence is increasing, if we start with ( a_1 < 1 ), then all subsequent ( a_n < 1 ), so ( x < 1 ), which is a subset of ( x < 2 ). Thus, the condition ( f(x) > x ) is satisfied for all ( x < 1 ), and since the sequence remains in ( x < 1 ), each subsequent term is greater than the previous one, and the entire sequence stays below 1. Therefore, starting from ( a_1 < 1 ), the sequence is strictly increasing and remains bounded above by 1.Alternatively, if we start with ( a_1 ) between 1 and 2, then ( a_2 = f(a_1) > 1 ), but ( a_2 = frac{1}{2 - a_1} ). Let's solve when ( a_2 > a_1 ).Given ( a_1 in (1, 2) ), ( a_2 = frac{1}{2 - a_1} ).We need ( a_2 > a_1 ):( frac{1}{2 - a_1} > a_1 )Multiply both sides by ( 2 - a_1 ), which is positive since ( a_1 < 2 ):( 1 > a_1(2 - a_1) )( 1 > 2a_1 - a_1^2 )( a_1^2 - 2a_1 + 1 > 0 )( (a_1 - 1)^2 > 0 )Which is true as long as ( a_1 neq 1 ). But since ( a_1 in (1, 2) ), this inequality holds. Therefore, even for ( a_1 in (1, 2) ), ( a_2 > a_1 ). However, as discussed earlier, this leads to ( a_2 > a_1 ), but ( a_2 = frac{1}{2 - a_1} ). Let's see where ( a_2 ) lies.If ( a_1 in (1, 2) ), then ( 2 - a_1 in (0, 1) ), so ( a_2 = frac{1}{2 - a_1} > 1 ). But as ( a_1 ) approaches 2 from below, ( a_2 ) approaches infinity. Therefore, if we start with ( a_1 in (1, 2) ), ( a_2 ) is greater than ( a_1 ), but ( a_2 ) can be much larger. However, the next term ( a_3 = frac{1}{2 - a_2} ). If ( a_2 > 2 ), then ( 2 - a_2 < 0 ), so ( a_3 ) is negative. Then, ( a_3 < 0 < a_2 ), which violates the increasing condition. Therefore, even though ( a_2 > a_1 ), ( a_3 ) would be less than ( a_2 ), making the sequence decrease at that point. Therefore, starting with ( a_1 in (1, 2) ) leads to a violation of the strictly increasing condition after two steps. Therefore, such starting values are invalid.Therefore, the only valid starting values are those where ( a_1 < 1 ), ensuring that all terms ( a_n < 1 ), and each subsequent term is greater than the previous one, approaching 1 asymptotically.Therefore, the range of ( a_1 ) is all real numbers less than 1.But wait, let's confirm this with another example. Suppose ( a_1 = 0.5 ).Then ( a_2 = 1/(2 - 0.5) = 1/1.5 ≈ 0.6667 )( a_3 = 1/(2 - 0.6667) ≈ 1/1.3333 ≈ 0.75 )( a_4 ≈ 1/(2 - 0.75) = 1/1.25 = 0.8 )( a_5 ≈ 1/(2 - 0.8) = 1/1.2 ≈ 0.8333 )Continuing this, the terms approach 1, always increasing. So this works.Another test: ( a_1 = -3 )( a_2 = 1/(2 - (-3)) = 1/5 = 0.2 )( a_3 = 1/(2 - 0.2) = 1/1.8 ≈ 0.5556 )( a_4 = 1/(2 - 0.5556) ≈ 1/1.4444 ≈ 0.6923 )( a_5 ≈ 1/(2 - 0.6923) ≈ 1/1.3077 ≈ 0.765 )And so on, increasing towards 1. So even negative ( a_1 ) work.Therefore, it seems that the only restriction on ( a_1 ) is that it must be less than 1. Therefore, the range of ( a_1 ) is ( (-infty, 1) ).But let me check once more for ( a_1 = 1 ). As before, ( a_1 = 1 ) leads to all terms being 1, which is not strictly increasing. So ( a_1 ) cannot be 1.If ( a_1 = 1 ), then ( a_2 = 1/(2 - 1) = 1 ), and so on. Not increasing.If ( a_1 = 2 ), undefined. If ( a_1 > 2 ), next term is negative, which violates increasing.Therefore, the conclusion is that ( a_1 ) must be less than 1.Therefore, the range of values for ( a_1 ) is all real numbers less than 1, so ( (-infty, 1) ).But wait, let me check if the problem is expecting an interval or some other constraints. The problem says "the range of values for ( a_1 )", so likely an interval. Since in Chinese problems like this, sometimes the answer is an interval like ( (-infty, 1) ), which would be written as ( a_1 < 1 ). However, perhaps there's a lower bound I missed?Wait, but when ( a_1 ) is very negative, say ( a_1 ) approaches negative infinity, the next term ( a_2 = 1/(2 - a_1) ) approaches 0 from the positive side, which is greater than ( a_1 ). Then, each subsequent term is positive and increasing towards 1. So even if ( a_1 ) is extremely negative, the sequence still increases towards 1. Therefore, there is no lower bound. Hence, the range is indeed ( (-infty, 1) ).But let's check one more example where ( a_1 ) is a fraction less than 1 but close to 1.Take ( a_1 = 0.9 ).( a_2 = 1/(2 - 0.9) = 1/1.1 ≈ 0.9091 ).( a_3 = 1/(2 - 0.9091) ≈ 1/1.0909 ≈ 0.9167 ).( a_4 ≈ 1/(2 - 0.9167) ≈ 1/1.0833 ≈ 0.9231 ).Continuing this, the terms approach 1, always increasing. So yes, starting from ( a_1 = 0.9 ), it works.Therefore, the answer should be ( a_1 < 1 ), or in interval notation, ( (-infty, 1) ).Final AnswerThe range of values for ( a_{1} ) is boxed{(-infty, 1)}.</think>Given the sequence ({a_n}) defined by (a_{n+1} = frac{1}{2 - a_n}), we need to determine the range of values for (a_1) such that the sequence is strictly increasing for all positive integers (n).1. Recurrence Relation Analysis: - The sequence is defined recursively, and we need (a_{n+1} > a_n) for all (n geq 1). - We start by analyzing the inequality (frac{1}{2 - a_n} > a_n).2. Solving the Inequality: - For (a_n < 2), multiplying both sides by (2 - a_n) (positive) gives: [ 1 > a_n(2 - a_n) implies a_n^2 - 2a_n + 1 > 0 implies (a_n - 1)^2 > 0 ] - This inequality holds if (a_n neq 1).3. Fixed Point Analysis: - The fixed point of the recurrence relation is found by solving (x = frac{1}{2 - x}), leading to (x = 1). - If the sequence reaches 1, it will remain constant, which contradicts the strictly increasing condition.4. Behavior of the Sequence: - If (a_1 < 1), then (a_2 = frac{1}{2 - a_1} < 1) and the sequence remains bounded above by 1, increasing towards 1. - If (a_1 > 1), subsequent terms can exceed 2, leading to undefined or negative values, violating the increasing condition.5. Verification with Examples: - Starting values (a_1 < 1) result in a sequence that is strictly increasing and approaches 1. - Negative values of (a_1) also result in a sequence that becomes positive and increases towards 1.6. Conclusion: - The sequence must be strictly increasing and remain bounded above by 1. This is satisfied if (a_1 < 1).Thus, the range of values for (a_1) is (boxed{(-infty, 1)}).]

🔑:(14) From the problem, we knowa_{n+1}-a_{n}=frac{1}{2-a_{n}}-a_{n}=frac{left(a_{n}-1right)^{2}}{2-a_{n}}>0 Leftrightarrow a_{n}<1 Leftrightarrow a_{n+1}=frac{1}{2-a_{n}}a_{n}. If a_{1}>1, then there exists a positive integer m, such that a_{1}>frac{m+1}{m}. In fact, if a_{n}>frac{p+1}{p}, then2-a_{n}<frac{p}{p-1} .Thus, from a_{1}>frac{m+1}{m}, we can get a_{m}>frac{2}{1}=2, which means a_{m+1}<a_{m}. Therefore, the range of a_{1} is a_{1}<1.

🧠:[<think>Okay, let's see. I need to find the magnitude of the vector a + b, given some conditions. Hmm. First, the vectors are a = (x, 1), b = (2, y), and c = (-2, 2). The problem says that a is perpendicular to c, and b is parallel to c. Alright, so I have to use these two conditions to find x and y, then compute a + b and find its magnitude.Starting with the first condition: a is perpendicular to c. When two vectors are perpendicular, their dot product is zero. So the dot product of a and c should be zero. Let me write that down.The dot product of a and c is (x)(-2) + (1)(2) = -2x + 2. Since they are perpendicular, this should equal zero. So:-2x + 2 = 0Solving for x, subtract 2 from both sides:-2x = -2Divide both sides by -2:x = 1Alright, so x is 1. That takes care of the first part. Now, the second condition is that b is parallel to c. If two vectors are parallel, one is a scalar multiple of the other. So vector b should be a scalar multiple of vector c. Let's write that out.Given b = (2, y) and c = (-2, 2), there exists some scalar k such that:(2, y) = k*(-2, 2) = (-2k, 2k)Therefore, the components must be equal. So:2 = -2kandy = 2kLet's solve the first equation for k:2 = -2kDivide both sides by -2:k = -1Now plug this value of k into the second equation to find y:y = 2*(-1) = -2So y is -2. Alright, so now we have x = 1 and y = -2. Let me confirm that these values satisfy the original conditions.First, check if a is perpendicular to c. a = (1, 1) and c = (-2, 2). The dot product is (1)(-2) + (1)(2) = -2 + 2 = 0. Yep, that works.Next, check if b is parallel to c. b = (2, -2) and c = (-2, 2). Let's see if b is a scalar multiple of c. If we take c and multiply by -1, we get (2, -2), which is exactly b. So yes, b is parallel to c. Great, the values check out.Now, compute a + b. a is (1, 1) and b is (2, -2). Adding them component-wise:(1 + 2, 1 + (-2)) = (3, -1)So the vector a + b is (3, -1). Now, we need the magnitude of this vector. The magnitude of a vector (u, v) is sqrt(u² + v²). Let's compute that.For (3, -1):Magnitude = sqrt(3² + (-1)²) = sqrt(9 + 1) = sqrt(10)So the magnitude is sqrt(10). Let me verify once again to make sure I didn't make any calculation mistakes.Wait, 3 squared is 9, (-1) squared is 1, sum is 10. Square root of 10 is correct. Yep, that seems right.Therefore, the answer should be sqrt(10), which in boxed form is boxed{sqrt{10}}.Final AnswerThe magnitude of (overrightarrow{a}+overrightarrow{b}) is boxed{sqrt{10}}.</think>Given vectors (overrightarrow{a} = (x, 1)), (overrightarrow{b} = (2, y)), and (overrightarrow{c} = (-2, 2)), we know that (overrightarrow{a} perp overrightarrow{c}) and (overrightarrow{b} parallel overrightarrow{c}). We need to find the magnitude of (overrightarrow{a} + overrightarrow{b}).First, since (overrightarrow{a}) is perpendicular to (overrightarrow{c}), their dot product is zero:[overrightarrow{a} cdot overrightarrow{c} = x(-2) + 1(2) = -2x + 2 = 0]Solving for (x):[-2x + 2 = 0 implies -2x = -2 implies x = 1]Next, since (overrightarrow{b}) is parallel to (overrightarrow{c}), (overrightarrow{b}) must be a scalar multiple of (overrightarrow{c}). Let (k) be the scalar such that:[overrightarrow{b} = koverrightarrow{c} implies (2, y) = k(-2, 2)]This gives us the equations:[2 = -2k quad text{and} quad y = 2k]Solving for (k):[2 = -2k implies k = -1]Substituting (k) into the equation for (y):[y = 2(-1) = -2]Now, substituting (x = 1) and (y = -2) into the vectors (overrightarrow{a}) and (overrightarrow{b}):[overrightarrow{a} = (1, 1) quad text{and} quad overrightarrow{b} = (2, -2)]Adding these vectors:[overrightarrow{a} + overrightarrow{b} = (1 + 2, 1 + (-2)) = (3, -1)]The magnitude of (overrightarrow{a} + overrightarrow{b}) is:[sqrt{3^2 + (-1)^2} = sqrt{9 + 1} = sqrt{10}]Thus, the magnitude of (overrightarrow{a} + overrightarrow{b}) is (boxed{sqrt{10}}).,<think>Alright, let me try to figure out how to solve this problem step by step. So, we have vectors a, b, and c, with coordinates given as a = (x, 1), b = (2, y), and c = (-2, 2). The conditions are that vector a is perpendicular to vector c, and vector b is parallel to vector c. We need to find the magnitude of the vector a + b. First, let me recall some vector properties. If two vectors are perpendicular, their dot product is zero. And if two vectors are parallel, one is a scalar multiple of the other. So, those are the two key concepts here. Starting with the first condition: vector a is perpendicular to vector c. So, the dot product of a and c should be zero. Let me write that out. The dot product of a and c is (x * -2) + (1 * 2) = -2x + 2. Since they are perpendicular, this equals zero. So, -2x + 2 = 0. Let me solve for x. Subtracting 2 from both sides: -2x = -2. Then divide both sides by -2: x = 1. So, x is 1. That's straightforward. Now, moving on to the second condition: vector b is parallel to vector c. If they are parallel, then vector b must be a scalar multiple of vector c. So, there exists some scalar k such that (2, y) = k * (-2, 2). Let me write that as equations. So, 2 = -2k and y = 2k. Let's solve for k first. From the first equation, 2 = -2k. Divide both sides by -2: k = -1. Then plug k into the second equation to find y. y = 2 * (-1) = -2. So, y is -2. Alright, so we found x = 1 and y = -2. Now we need to find the magnitude of vector a + vector b. Let's first compute vector a + vector b. Vector a is (x, 1) which is (1, 1) since x =1. Vector b is (2, y) which is (2, -2) since y = -2. Adding these two vectors: (1 + 2, 1 + (-2)) = (3, -1). Now, the magnitude of this resultant vector (3, -1) is calculated by the square root of the sum of the squares of its components. So, magnitude = sqrt(3^2 + (-1)^2) = sqrt(9 + 1) = sqrt(10). Wait a second, let me verify each step again to make sure I didn't make a mistake. First, for the perpendicular condition: a · c = (x, 1) · (-2, 2) = -2x + 2. Setting that equal to zero gives -2x + 2 = 0, which leads to x = 1. That seems correct. For the parallel condition: vector b must be a scalar multiple of vector c. So, (2, y) = k*(-2, 2). Solving for k in the x-component: 2 = -2k => k = -1. Then y = 2k = 2*(-1) = -2. That checks out. Adding vectors a and b: a = (1, 1), b = (2, -2). Adding component-wise: (1+2, 1 + (-2)) = (3, -1). The magnitude is sqrt(3² + (-1)²) = sqrt(9 +1) = sqrt(10). Hmm, sqrt(10) is approximately 3.162, but the problem might want an exact answer, which is sqrt(10). Let me check if the question specifies the form. It says "find the magnitude," and they probably want the exact value, so sqrt(10) is the correct answer. Wait, just to be thorough, let me recheck the vector addition. If a is (1,1) and b is (2, -2), adding them:First component: 1 + 2 = 3.Second component: 1 + (-2) = -1.So, the sum vector is indeed (3, -1). The magnitude squared is 3² + (-1)² = 9 + 1 = 10, so the magnitude is sqrt(10). I think that's right. Let me just recap everything:- Found x by setting the dot product of a and c to zero.- Found y by setting b as a scalar multiple of c.- Added the vectors a and b.- Calculated the magnitude. All steps seem logically consistent and the arithmetic checks out. Just to make sure, let me consider if there's another way vectors could be parallel. But nope, if vectors are parallel, one is a scalar multiple of the other, which was applied correctly here. And perpendicular vectors have a dot product of zero, which was used properly. Therefore, I'm confident that the magnitude of vector a + vector b is sqrt(10).Final AnswerThe magnitude of (overrightarrow{a}+overrightarrow{b}) is boxed{sqrt{10}}).</think>Given vectors (overrightarrow{a} = (x, 1)), (overrightarrow{b} = (2, y)), and (overrightarrow{c} = (-2, 2)), we know that (overrightarrow{a} perp overrightarrow{c}) and (overrightarrow{b} parallel overrightarrow{c}). We need to find the magnitude of (overrightarrow{a} + overrightarrow{b}).1. Perpendicular Condition ((overrightarrow{a} perp overrightarrow{c})): - The dot product of (overrightarrow{a}) and (overrightarrow{c}) must be zero: [ overrightarrow{a} cdot overrightarrow{c} = (x)(-2) + (1)(2) = -2x + 2 = 0 ] - Solving for (x): [ -2x + 2 = 0 implies -2x = -2 implies x = 1 ]2. Parallel Condition ((overrightarrow{b} parallel overrightarrow{c})): - (overrightarrow{b}) must be a scalar multiple of (overrightarrow{c}): [ overrightarrow{b} = k overrightarrow{c} implies (2, y) = k(-2, 2) ] - Solving for (k) using the components: [ 2 = -2k implies k = -1 ] [ y = 2k = 2(-1) = -2 ]3. Vector Addition: - Substituting (x = 1) and (y = -2): [ overrightarrow{a} = (1, 1), quad overrightarrow{b} = (2, -2) ] - Adding the vectors: [ overrightarrow{a} + overrightarrow{b} = (1 + 2, 1 + (-2)) = (3, -1) ]4. Magnitude Calculation: - The magnitude of (overrightarrow{a} + overrightarrow{b}) is: [ | overrightarrow{a} + overrightarrow{b} | = sqrt{3^2 + (-1)^2} = sqrt{9 + 1} = sqrt{10} ]The magnitude of (overrightarrow{a} + overrightarrow{b}) is (boxed{sqrt{10}}).]

🔑:AnalysisThis problem examines the representation and operation of parallel and perpendicular vectors in a plane, as well as the problem of finding the magnitude of a vector, which is a basic topic.SolutionGiven vectors (overrightarrow{a}=(x,1)), (overrightarrow{b}=(2,y)), (overrightarrow{c}=(-2,2)), and (overrightarrow{a} perp overrightarrow{c}), (overrightarrow{b} parallel overrightarrow{c}),(begin{cases}-2x+2=0 4+2y=0end{cases});Solving these equations, we find (x=1) and (y=-2),Therefore, (overrightarrow{a}=(1,1)) and (overrightarrow{b}=(2,-2));Thus, (overrightarrow{a}+overrightarrow{b}=(3,-1)),The magnitude of (overrightarrow{a}+overrightarrow{b}) is (sqrt{3^2+(-1)^2} = sqrt{10}).Hence, the answer is (boxed{sqrt{10}}).

🧠:[<think>Okay, let's see. I need to find the equation of a line that passes through the point P(6,7) and is tangent to the hyperbola x²/9 - y²/16 = 1. Hmm, tangent lines to hyperbolas... I remember there's a standard formula for the tangent to a hyperbola, similar to the one for ellipses. Let me recall. For a hyperbola of the form x²/a² - y²/b² = 1, the equation of the tangent line at a point (x₁, y₁) on the hyperbola is (xx₁)/a² - (yy₁)/b² = 1. But wait, that's if the point is on the hyperbola. However, in this problem, the point P(6,7) might not be on the hyperbola. So, I can't directly use that formula. Hmm.Alternatively, maybe I can use the condition for a line to be tangent to a hyperbola. If I have a general line y = mx + c, then substituting this into the hyperbola equation should result in a quadratic equation with exactly one solution (since the line is tangent). That would mean the discriminant of the quadratic is zero. That might be a way to approach it. Let's try that.First, let's assume the equation of the tangent line is y = mx + c. Since it passes through the point P(6,7), substituting x=6 and y=7 into the equation gives 7 = 6m + c. So, c = 7 - 6m. Therefore, the equation of the line can be written as y = mx + (7 - 6m). Now, substitute this into the hyperbola equation and set the discriminant to zero.The hyperbola equation is x²/9 - y²/16 = 1. Substitute y = mx + (7 - 6m):x²/9 - [mx + (7 - 6m)]²/16 = 1.Let me expand the numerator in the second term:[mx + (7 - 6m)]² = m²x² + 2m(7 - 6m)x + (7 - 6m)².So, substituting back into the equation:x²/9 - [m²x² + 2m(7 - 6m)x + (7 - 6m)²]/16 = 1.Multiply through by 144 (the least common multiple of 9 and 16) to eliminate denominators:16x² - 9[m²x² + 2m(7 - 6m)x + (7 - 6m)²] = 144.Expanding this:16x² - 9m²x² - 18m(7 - 6m)x - 9(7 - 6m)² = 144.Combine like terms:(16 - 9m²)x² - [18m(7 - 6m)]x - [9(7 - 6m)² + 144] = 0.This is a quadratic in x. For the line to be tangent to the hyperbola, the discriminant of this quadratic must be zero. The discriminant D is given by:D = [ -18m(7 - 6m) ]² - 4*(16 - 9m²)*[ -9(7 - 6m)² - 144 ].Wait, that seems complicated. Let me check each term step by step. The quadratic is Ax² + Bx + C = 0, where:A = 16 - 9m²,B = -18m(7 - 6m),C = -9(7 - 6m)² - 144.So, discriminant D = B² - 4AC.Therefore, D = [ -18m(7 - 6m) ]² - 4*(16 - 9m²)*[ -9(7 - 6m)² - 144 ].Let me compute each part:First, B²:[ -18m(7 - 6m) ]² = [18m(7 - 6m)]² = (18)^2 * m² * (7 - 6m)^2 = 324m²(7 - 6m)^2.Then, 4AC:4*(16 - 9m²)*[ -9(7 - 6m)^2 - 144 ].Let me compute the term inside the brackets first:-9(7 - 6m)^2 - 144 = -9[(7 - 6m)^2 + 16].Wait, maybe factor out -9:= -9[(7 - 6m)^2 + 16].But perhaps expanding (7 - 6m)^2 first would help. Let's do that:(7 - 6m)^2 = 49 - 84m + 36m².So,-9(49 - 84m + 36m²) - 144 = -9*49 + 9*84m - 9*36m² - 144Calculate each term:-9*49 = -441,9*84m = 756m,-9*36m² = -324m²,Then -441 - 144 = -585.So, combining these:-441 + 756m - 324m² -144 = -585 + 756m - 324m².Therefore, the term inside the brackets simplifies to -324m² + 756m - 585.Therefore, 4AC becomes:4*(16 - 9m²)*(-324m² + 756m - 585).So, putting it all together:D = 324m²(7 - 6m)^2 - 4*(16 - 9m²)*(-324m² + 756m - 585).This is going to be a bit messy, but let's try to compute each term step by step.First, compute 324m²(7 - 6m)^2:We already know that (7 - 6m)^2 = 49 - 84m + 36m², so:324m²*(49 - 84m + 36m²) = 324m²*49 - 324m²*84m + 324m²*36m²Compute each term:324*49 = 15876,So, 15876m².324*84 = let's compute 300*84 = 25200, 24*84 = 2016, total 25200 + 2016 = 27216. So, 27216m³.324*36 = 11664, so 11664m^4.Therefore, 324m²(7 - 6m)^2 = 11664m^4 - 27216m³ + 15876m².Now, compute the second term: -4*(16 - 9m²)*(-324m² + 756m - 585).First, compute (16 - 9m²)*(-324m² + 756m - 585):Multiply each term:16*(-324m²) = -5184m²,16*756m = 12096m,16*(-585) = -9360,-9m²*(-324m²) = 2916m^4,-9m²*756m = -6804m³,-9m²*(-585) = 5265m².So, adding all these:2916m^4 - 6804m³ + ( -5184m² + 5265m² ) + 12096m - 9360Simplify:2916m^4 - 6804m³ + (5265 - 5184)m² + 12096m - 9360Which is:2916m^4 - 6804m³ + 81m² + 12096m - 9360Now, multiply by -4:-4*(2916m^4 - 6804m³ + 81m² + 12096m - 9360) =-11664m^4 + 27216m³ - 324m² - 48384m + 37440So now, putting all together into discriminant D:D = [11664m^4 - 27216m³ + 15876m²] + [ -11664m^4 + 27216m³ - 324m² - 48384m + 37440 ]Let's combine term by term:11664m^4 - 11664m^4 = 0,-27216m³ + 27216m³ = 0,15876m² - 324m² = 15552m²,-48384m,+37440.So, D = 15552m² - 48384m + 37440.Set discriminant D to zero:15552m² - 48384m + 37440 = 0.Hmm, that's a quadratic in m. Let's try to simplify this equation.First, notice that all coefficients are divisible by 144:15552 ÷ 144 = 108,48384 ÷ 144 = 336,37440 ÷ 144 = 260.So, divide both sides by 144:108m² - 336m + 260 = 0.Wait, let's check:144*108 = 15552,144*336 = 144*300 + 144*36 = 43200 + 5184 = 48384,144*260 = 144*200 + 144*60 = 28800 + 8640 = 37440. Correct.So, simplified equation is 108m² - 336m + 260 = 0.Can we simplify further? Let's check if there is a common factor.108, 336, 260: all divisible by 4? Let's see:108 ÷ 4 = 27,336 ÷ 4 = 84,260 ÷ 4 = 65.So, 27m² - 84m + 65 = 0.Hmm, does this quadratic factor?Looking for two numbers a and b such that a*b = 27*65 = 1755 and a + b = -84.Wait, maybe use quadratic formula.Quadratic equation: 27m² -84m +65=0Compute discriminant D' = (-84)^2 -4*27*65Which is 7056 - 4*27*65First compute 4*27=108, then 108*65=7020So D' = 7056 - 7020 = 36Therefore, sqrt(36)=6Thus, solutions are:m = [84 ±6]/(2*27) = (84 ±6)/54Therefore,m = (84 +6)/54 =90/54=5/3 ≈1.6667,or m=(84 -6)/54=78/54=13/9≈1.4444.So the two possible slopes are m=5/3 and m=13/9.Therefore, the equations of the tangent lines can be found using these slopes and the point P(6,7). Recall that the equation of the line is y= mx + (7 -6m). So plugging in each m:First line, m=5/3:c =7 -6*(5/3)=7 -10= -3.So equation is y= (5/3)x -3.Multiply both sides by 3 to eliminate fraction: 3y=5x -9 =>5x -3y -9=0.Second line, m=13/9:c=7 -6*(13/9)=7 - (78/9)=7 -26/3= (21/3 -26/3)= -5/3.So equation is y=(13/9)x -5/3.Multiply both sides by 9: 9y=13x -15 =>13x -9y -15=0.Therefore, the two tangent lines are 5x -3y -9=0 and 13x -9y -15=0.Wait, but let me check if these lines are indeed tangent to the hyperbola and pass through (6,7).First, check if they pass through (6,7):For 5x -3y -9=0:5*6 -3*7 -9=30 -21 -9=0. Correct.For 13x -9y -15=0:13*6 -9*7 -15=78 -63 -15=0. Correct.Now, check if they are tangent to the hyperbola. Let's take the first line, 5x -3y -9=0. Let's write it as y= (5/3)x -3.Substitute into hyperbola:x²/9 - [(5/3 x -3)^2]/16=1.Compute (5/3 x -3)^2:= (25/9)x² - 10x +9.So,x²/9 - [25/9 x² -10x +9]/16=1.Multiply through by 144:16x² -9*(25x² -90x +81)=144.16x² -225x² +810x -729=144.-209x² +810x -729 -144=0.-209x² +810x -873=0.Multiply by -1:209x² -810x +873=0.Compute discriminant D:(810)^2 -4*209*873.Compute 810^2=656100,4*209=836,836*873.Let me compute 800*873=698,400,36*873=31,428,Total=698,400+31,428=729,828.Thus D=656,100 -729,828= -73,728.Wait, that can't be. The discriminant is negative? That would mean no intersection points, but the line is supposed to be tangent. Something's wrong here.Wait, I must have made an error in substitution or calculation. Let me check the substitution again.Original hyperbola: x²/9 - y²/16 =1.Line: 5x -3y =9 => y= (5x -9)/3.So substitute into hyperbola:x²/9 - [ ( (5x -9)/3 )² ] /16 =1.Compute (5x -9)² =25x² -90x +81.Divide by 9: (25x² -90x +81)/9.Then divide by 16: (25x² -90x +81)/(9*16)= (25x² -90x +81)/144.Thus, equation becomes:x²/9 - (25x² -90x +81)/144 =1.Multiply through by 144:16x² - (25x² -90x +81) =144.16x² -25x² +90x -81 -144=0.-9x² +90x -225=0.Multiply by -1:9x² -90x +225=0.Divide by 9:x² -10x +25=0.Which factors to (x -5)^2=0, so x=5. So discriminant is zero, as it should be. So x=5 is the point of contact. Then y=(5*5 -9)/3=(25 -9)/3=16/3. So the tangent point is (5,16/3). Therefore, it is indeed tangent. My previous calculation had a mistake in expanding (5/3 x -3)^2. I think I miscalculated the cross term. Let's see:(5/3 x -3)^2 = (5/3 x)^2 - 2*(5/3 x)*3 + 3^2 = 25/9 x² - 10x +9. Wait, so then when I substituted into the hyperbola equation, I should have:x²/9 - [25/9 x² -10x +9]/16 =1.Multiply by 144:16x² -9*(25x² -90x +81) =144.Wait, here 25x² -90x +81 is correct? Because [25/9 x² -10x +9] multiplied by 9 gives 25x² -90x +81. Yes. So then:16x² -225x² +810x -729=144.Which is -209x² +810x -873=0. Wait, but earlier substitution with the line equation y=(5x-9)/3 gave x² -10x +25=0. There's a discrepancy here. Which one is correct?Wait, when I used substitution of y=(5x -9)/3 into the hyperbola, I correctly arrived at (x -5)^2=0. So there's an error in the earlier method where I directly substituted y=mx +c into the hyperbola equation. What did I do wrong there?Wait, perhaps in the initial substitution, when I used y=mx + c and computed the discriminant, leading to m=5/3 and m=13/9, but when I check the line with m=5/3, substituting back gives a quadratic with a discriminant of zero, but when I did the substitution step by step, there was a miscalculation. Wait, perhaps the mistake was in the discriminant calculation.Wait, let's retrace. When I set up the equation after substituting y = mx + (7 -6m) into the hyperbola, I multiplied through by 144 and arrived at the quadratic equation:(16 -9m²)x² -18m(7 -6m)x -[9(7 -6m)^2 +144]=0.Then, the discriminant D = [ -18m(7 -6m) ]² -4*(16 -9m²)*(-9(7 -6m)^2 -144).But when I computed the term C, which was -9(7 -6m)^2 -144, but in the quadratic equation, the constant term is C = -9(7 -6m)^2 -144.However, when I substituted m=5/3 into this, the discriminant turned out to be zero. But when I calculated earlier using the line equation, the discriminant was zero. So where did the mistake happen in the intermediate steps?Wait, maybe when I expanded D, the discriminant, in terms of m, I made an error. Let me check again.Original discriminant D = 324m²(7 -6m)^2 -4*(16 -9m²)*(-9(7 -6m)^2 -144).But when I expanded this, I ended up with D=15552m² -48384m +37440=0. Solving this gave m=5/3 and m=13/9.But when I substituted m=5/3 into the discriminant D expression, which should be zero, but when I checked with the line y=(5/3)x -3, substituting into hyperbola gave a discriminant of zero. So why when I did the substitution earlier, expanding the discriminant D, I had conflicting results?Wait, perhaps when I expanded D, I made a miscalculation. Let me check the step where I combined the terms.After expanding both parts:First part: 324m²(7 -6m)^2 = 11664m^4 -27216m³ +15876m².Second part: -4*(16 -9m²)*(-324m² +756m -585)= -11664m^4 +27216m³ -324m² -48384m +37440.Then, combining these:11664m^4 -27216m³ +15876m² -11664m^4 +27216m³ -324m² -48384m +37440.Indeed, as I did before:11664m^4 -11664m^4 =0,-27216m³ +27216m³=0,15876m² -324m²=15552m²,-48384m,+37440.Therefore, D=15552m² -48384m +37440=0.Divide by 144: 108m² -336m +260=0,Then divide by 4:27m² -84m +65=0,Which had solutions m=5/3 and m=13/9. So these slopes are correct.But when I checked m=5/3 with the line y= (5/3)x -3, substituting into hyperbola gave a quadratic equation with discriminant zero. So why, when I did the substitution via expanding the discriminant in terms of m, I arrived at a different quadratic equation? Wait, maybe when I substituted the line into the hyperbola equation, I made a mistake in arithmetic.Wait, let's do this again. For the line y=(5/3)x -3, substituting into the hyperbola:x²/9 - y²/16 =1.So y=(5x/3 -3), so y²=(25x²/9 -10x +9).Thus, substituting:x²/9 - (25x²/9 -10x +9)/16 =1.Multiply through by 144 (LCM of 9 and 16):16x² -9*(25x² -90x +81)=144.Compute 9*(25x² -90x +81)=225x² -810x +729.Thus,16x² -225x² +810x -729=144.Combine terms:-209x² +810x -729 -144=0 => -209x² +810x -873=0.Multiply by -1:209x² -810x +873=0.Compute discriminant:D= (-810)^2 -4*209*873.Calculate:810^2=656100,4*209=836,836*873.Let me compute 800*873=698,400,36*873=31,428,Total=698,400 +31,428=729,828.So D=656,100 -729,828= -73,728.Wait, this is negative? That can't be. But when I did the substitution earlier, I got a perfect square. There's a contradiction here.Wait, I think when I substitute y=(5x -9)/3 into the hyperbola, I get:x²/9 - [(5x -9)/3]^2 /16 =1.Compute [(5x -9)/3]^2 = (25x² -90x +81)/9.Thus,x²/9 - (25x² -90x +81)/(9*16) =1.Multiply through by 144:16x² - (25x² -90x +81) =144.16x² -25x² +90x -81=144.-9x² +90x -81 -144=0.-9x² +90x -225=0.Multiply by -1:9x² -90x +225=0.Divide by 9:x² -10x +25=0.Which is (x-5)^2=0. So discriminant is zero. Therefore, this is correct. So why does substituting into the line y=(5/3)x -3 give a different equation?Wait, perhaps I messed up the line equation. Wait, 5x -3y -9=0 is equivalent to 3y=5x -9, so y=(5/3)x -3. So that's correct. Then substituting into hyperbola gives (x-5)^2=0, so x=5, y=(25/3 -9)= (25 -27)/3= (-2)/3? Wait, no.Wait, if x=5, then y=(5*5 -9)/3=(25 -9)/3=16/3≈5.333. So the point of tangency is (5,16/3). Let's check if this point is on the hyperbola:x²/9 - y²/16 =25/9 - (256/9)/16=25/9 -16/9=9/9=1. Correct. So it is on the hyperbola, and the line is tangent at that point. Therefore, the line is indeed tangent.But when I substituted y=(5/3)x -3 into the hyperbola equation, I arrived at -209x² +810x -873=0, which had discriminant D= -73,728. That seems contradictory. Wait, no. Wait, in the first substitution method, after multiplying through by 144, we get 16x² -225x² +810x -729=144, which leads to -209x² +810x -873=0. But this is incorrect, because when I do the substitution correctly, step by step, I get x² -10x +25=0. How is that possible?Wait, maybe I made a mistake in the initial substitution with the general line equation. Let me check.Original problem: substitute y = mx + c into hyperbola x²/9 - y²/16 =1, with c=7 -6m. Then after substitution and simplifying, we get a quadratic equation in x. The discriminant of that quadratic must be zero. Then solving for m gives m=5/3 and m=13/9.But when we check m=5/3, we found that substituting the specific line equation into the hyperbola gives a perfect square, hence discriminant zero. However, when I substituted into the general discriminant expression, I got a different equation. Wait, no. Wait, maybe there was a miscalculation when expanding the general discriminant.Wait, let me check the quadratic equation after substituting the general line y=mx + (7 -6m) into the hyperbola:After substituting and multiplying by 144, we get:16x² -9[m²x² + 2m(7 -6m)x + (7 -6m)^2] =144.Expanding:16x² -9m²x² -18m(7 -6m)x -9(7 -6m)^2 =144.Then moving 144 to the left:16x² -9m²x² -18m(7 -6m)x -9(7 -6m)^2 -144=0.Factor terms:(16 -9m²)x² -18m(7 -6m)x -[9(7 -6m)^2 +144]=0.So A=16 -9m²,B= -18m(7 -6m),C= -9(7 -6m)^2 -144.Thus discriminant D=B² -4AC.Compute D:[ -18m(7 -6m) ]² -4*(16 -9m²)*[ -9(7 -6m)^2 -144 ]=324m²(7 -6m)^2 -4*(16 -9m²)*(-9(7 -6m)^2 -144)Compute term by term:First term:324m²(7 -6m)^2.Second term: -4*(16 -9m²)*(-9(7 -6m)^2 -144).Let's compute the second term:First, inside the brackets: -9(7 -6m)^2 -144 = -9[(7 -6m)^2 +16].But expanding (7 -6m)^2:=49 -84m +36m².Thus, -9[49 -84m +36m² +16] = -9[65 -84m +36m²] = -585 +756m -324m².So the second term becomes:-4*(16 -9m²)*(-324m² +756m -585)= -4*[16*(-324m² +756m -585) -9m²*(-324m² +756m -585)]= -4*[ -5184m² +12096m -9360 +2916m^4 -6804m³ +5265m² ]= -4*[2916m^4 -6804m³ + ( -5184m² +5265m² ) +12096m -9360 ]= -4*[2916m^4 -6804m³ +81m² +12096m -9360 ]= -11664m^4 +27216m³ -324m² -48384m +37440.Then, adding the first term:First term:324m²(7 -6m)^2=324m²*(49 -84m +36m²)=324*49m² -324*84m³ +324*36m^4=15876m² -27216m³ +11664m^4.So total D=11664m^4 -27216m³ +15876m² -11664m^4 +27216m³ -324m² -48384m +37440=15552m² -48384m +37440.Set equal to zero:15552m² -48384m +37440=0.Divide by 144:108m² -336m +260=0.Divide by 4:27m² -84m +65=0.Solutions m=(84±√(84² -4*27*65))/54=(84±√(7056-7020))/54=(84±√36)/54=(84±6)/54.Thus m=90/54=5/3 or m=78/54=13/9.Therefore, the calculation seems correct. Then why, when substituting m=5/3 into the quadratic equation, we get a different discriminant?Wait, let's substitute m=5/3 into the quadratic equation:A=16 -9*(25/9)=16 -25= -9,B= -18*(5/3)*(7 -6*(5/3))= -18*(5/3)*(7 -10)= -18*(5/3)*(-3)= -18*(-5)=90,C= -9*(7 -6*(5/3))² -144= -9*(7 -10)^2 -144= -9*(9) -144= -81 -144= -225.So quadratic equation is -9x² +90x -225=0, which simplifies to -9x² +90x -225=0, divide by -9: x² -10x +25=0, discriminant D=100 -100=0. Correct.Therefore, discriminant D is zero. So the error was in my initial substitution where I used y=(5/3)x -3 and arrived at a different quadratic. Wait, no, in that case, I think I must have messed up the multiplication by 144.Wait, in the first substitution, when I substituted y=(5/3)x -3 into the hyperbola equation, I had:x²/9 - [(5/3 x -3)^2]/16=1.Multiply by 144:16x² -9*(25x²/9 -10x +9)=144.But expanding this:16x² - (225x²/9 -90x +81)=144.Wait, 9*(25x²/9 -10x +9)=25x² -90x +81.So 16x² -25x² +90x -81=144.Thus, -9x² +90x -225=0.Which is correct, and discriminant zero.Earlier, when I tried expanding the general discriminant, I had a different coefficient for x², but that was in terms of m. When substituting m=5/3 into the general discriminant, it indeed becomes zero, as shown.Therefore, the problem was in my initial mistaken substitution where I thought the discriminant was negative, but I must have made an arithmetic error. The correct calculation shows that for m=5/3, the discriminant is zero. Similarly for m=13/9.Thus, the two lines are indeed tangent to the hyperbola and pass through (6,7).Therefore, the equations are:For m=5/3: y= (5/3)x -3, which can be written as 5x -3y -9=0,For m=13/9: y=(13/9)x -5/3, which can be written as 13x -9y -15=0.To confirm, let's check the second line:13x -9y -15=0.At point (6,7):13*6 -9*7 -15=78 -63 -15=0. Correct.Now, check if this line is tangent to the hyperbola.Write it as y=(13/9)x -5/3.Substitute into hyperbola:x²/9 - [(13/9 x -5/3)^2]/16=1.Compute (13/9 x -5/3)^2:= (169/81)x² - (130/27)x +25/9.Thus,x²/9 - [169/81 x² -130/27 x +25/9]/16=1.Multiply through by 144:16x² -9*(169x² -390x +225)/81=144.Simplify:16x² - (169x² -390x +225)/9=144.Multiply numerator and denominator:16x² -169x²/9 +390x/9 -225/9=144.Multiply everything by 9 to eliminate denominators:144x² -169x² +390x -225=1296.Combine terms:(144x² -169x²)= -25x²,So:-25x² +390x -225 -1296=0,-25x² +390x -1521=0,Multiply by -1:25x² -390x +1521=0.Compute discriminant:D=(-390)^2 -4*25*1521.Calculate:390^2=152,100,4*25=100,100*1521=152,100.Thus D=152,100 -152,100=0.So discriminant is zero. Therefore, this line is also tangent to the hyperbola. The point of tangency can be found by solving 25x² -390x +1521=0.Which is x=(390 ±0)/50=390/50=7.8=39/5. Then y=(13/9)(39/5) -5/3= (507/45) -75/45=432/45=48/5. So the point is (39/5, 48/5). Check if this is on the hyperbola:x²/9 - y²/16=( (1521/25)/9 ) - ( 2304/25 /16 )=(1521/225) - (2304/400)= (1521*16 -2304*9)/3600=(24336 -20736)/3600=3600/3600=1. Correct.Therefore, both lines are indeed tangent to the hyperbola and pass through (6,7).Hence, the equations of the tangent lines are 5x -3y -9=0 and 13x -9y -15=0.Final AnswerThe equations of the tangent lines are boxed{5x - 3y - 9 = 0} and boxed{13x - 9y - 15 = 0}.</think>To find the equation of the line passing through the point ( P(6, 7) ) and tangent to the hyperbola ( frac{x^2}{9} - frac{y^2}{16} = 1 ), we start by assuming the equation of the tangent line is ( y = mx + c ). Since the line passes through ( P(6, 7) ), we have ( 7 = 6m + c ), giving ( c = 7 - 6m ). Thus, the line equation is ( y = mx + (7 - 6m) ).Substituting ( y = mx + (7 - 6m) ) into the hyperbola equation:[frac{x^2}{9} - frac{(mx + (7 - 6m))^2}{16} = 1]Expanding and simplifying this equation, we get a quadratic in ( x ):[left(16 - 9m^2right)x^2 - 18m(7 - 6m)x - left[9(7 - 6m)^2 + 144right] = 0]For the line to be tangent, the discriminant of this quadratic must be zero. The discriminant ( D ) is given by:[D = left[-18m(7 - 6m)right]^2 - 4left(16 - 9m^2right)left[-9(7 - 6m)^2 - 144right]]Simplifying the discriminant, we find:[15552m^2 - 48384m + 37440 = 0]Dividing by 144 and simplifying further:[27m^2 - 84m + 65 = 0]Solving this quadratic equation using the quadratic formula, we get the slopes ( m = frac{5}{3} ) and ( m = frac{13}{9} ).For ( m = frac{5}{3} ), the intercept ( c ) is:[c = 7 - 6 left(frac{5}{3}right) = 7 - 10 = -3]Thus, the equation of the line is ( y = frac{5}{3}x - 3 ), or ( 5x - 3y - 9 = 0 ).For ( m = frac{13}{9} ), the intercept ( c ) is:[c = 7 - 6 left(frac{13}{9}right) = 7 - frac{78}{9} = -frac{5}{3}]Thus, the equation of the line is ( y = frac{13}{9}x - frac{5}{3} ), or ( 13x - 9y - 15 = 0 ).Both lines pass through ( P(6, 7) ) and are tangent to the hyperbola. The equations of the tangent lines are:[boxed{5x - 3y - 9 = 0} quad text{and} quad boxed{13x - 9y - 15 = 0}],<think>Okay, so I need to find the equation of a line that passes through the point P(6,7) and is tangent to the hyperbola x²/9 - y²/16 = 1. Hmm, let me start by recalling what a tangent line to a hyperbola is like. First, the general equation of a hyperbola is x²/a² - y²/b² = 1, and in this case, a² is 9 and b² is 16, so a = 3 and b = 4. The hyperbola is centered at the origin and opens left and right along the x-axis.Now, tangent lines to hyperbolas have a specific form. For a hyperbola x²/a² - y²/b² = 1, the equation of the tangent line at a point (x₀, y₀) on the hyperbola is (xx₀)/a² - (yy₀)/b² = 1. But wait, the problem doesn't give a specific point of tangency; instead, it gives a point through which the tangent line passes. So, I can't directly use that formula. Maybe I need another approach.Another way to find the tangent line is to use the condition that a line is tangent to a hyperbola if it intersects the hyperbola at exactly one point. So, if I can write the equation of a line passing through (6,7) and then find the condition that when solving the system of the line and the hyperbola, there's only one solution. That should give me the equation(s) of the tangent line(s).Let me try that approach. Let's denote the equation of the line as y = mx + c. Since it passes through (6,7), substituting x = 6 and y = 7 gives 7 = 6m + c. So, c = 7 - 6m. Therefore, the equation of the line can be written as y = mx + (7 - 6m).Now, substitute this into the hyperbola equation:x²/9 - (mx + 7 - 6m)²/16 = 1Expanding this:x²/9 - [m²x² + 2m(7 - 6m)x + (7 - 6m)²]/16 = 1Multiply out each term:Let me compute each part step by step. First, expand the square in the numerator:(mx + (7 - 6m))² = m²x² + 2m(7 - 6m)x + (7 - 6m)²So substituting back into the hyperbola equation:x²/9 - [m²x² + 14m x - 12m²x + 49 - 84m + 36m²]/16 = 1Wait, actually, let me check that expansion again. Let me do the expansion properly:(mx + (7 - 6m))² = m²x² + 2 * mx * (7 - 6m) + (7 - 6m)^2= m²x² + (14m - 12m²)x + (49 - 84m + 36m²)Yes, that's correct.So now plugging that back into the hyperbola equation:x²/9 - [m²x² + (14m - 12m²)x + 49 - 84m + 36m²]/16 = 1Now, let's combine the terms:Multiply each term by 144 (the least common multiple of 9 and 16) to eliminate denominators:144*(x²/9) - 144*[m²x² + (14m - 12m²)x + 49 - 84m + 36m²]/16 = 144*1Calculating each term:144*(x²/9) = 16x²144*[m²x² + (14m - 12m²)x + 49 - 84m + 36m²]/16 = 9*(m²x² + (14m - 12m²)x + 49 - 84m + 36m²)And the right-hand side is 144.So the equation becomes:16x² - 9[m²x² + (14m - 12m²)x + 49 - 84m + 36m²] = 144Let me expand this:16x² - 9m²x² - 9*(14m - 12m²)x - 9*(49 - 84m + 36m²) = 144Simplify each term:First term: 16x² - 9m²x² = x²(16 - 9m²)Second term: -9*(14m - 12m²)x = -126m x + 108m² xThird term: -9*(49 - 84m + 36m²) = -441 + 756m - 324m²So putting it all together:x²(16 - 9m²) + (-126m + 108m²)x + (-441 + 756m - 324m²) - 144 = 0Wait, the right-hand side was 144, so when moving it to the left, subtract 144:So combining the constants:-441 - 144 = -585Therefore, the equation becomes:(16 - 9m²)x² + (108m² - 126m)x + (-585 + 756m - 324m²) = 0Now, for this quadratic equation in x to have exactly one solution, the discriminant must be zero. So discriminant D = [B]^2 - 4AC = 0, where:A = (16 - 9m²)B = (108m² - 126m)C = (-585 + 756m - 324m²)Let me compute D:D = (108m² - 126m)^2 - 4*(16 - 9m²)*(-585 + 756m - 324m²)This looks complicated. Let me compute each part step by step.First, compute (108m² - 126m)^2:Let me factor out 18m from 108m² - 126m:108m² - 126m = 18m(6m - 7)But maybe squaring directly is easier.(108m² - 126m)^2 = (108m²)^2 - 2*108m²*126m + (126m)^2Compute each term:108^2 = 11664, so (108m²)^2 = 11664m^42*108*126 = 2*13608 = 27216, so the second term is -27216m^3126^2 = 15876, so (126m)^2 = 15876m²Thus, (108m² - 126m)^2 = 11664m^4 - 27216m^3 + 15876m²Now, compute the second part: -4*(16 - 9m²)*(-585 + 756m - 324m²)First, compute the product (16 - 9m²)(-585 + 756m - 324m²)Let me distribute:16*(-585) + 16*756m + 16*(-324m²) -9m²*(-585) + (-9m²)*756m + (-9m²)*(-324m²)Compute each term:16*(-585) = -936016*756m = 12096m16*(-324m²) = -5184m²-9m²*(-585) = 5265m²-9m²*756m = -6804m³-9m²*(-324m²) = 2916m^4Now, combine all these terms:-9360 + 12096m -5184m² + 5265m² -6804m³ + 2916m^4Combine like terms:For m^4: 2916m^4For m^3: -6804m³For m²: (-5184 + 5265)m² = 81m²For m: 12096mConstants: -9360So the product is 2916m^4 -6804m³ +81m² +12096m -9360Multiply this by -4:-4*(2916m^4 -6804m³ +81m² +12096m -9360) = -11664m^4 +27216m³ -324m² -48384m +37440Therefore, the discriminant D is:[11664m^4 -27216m^3 +15876m²] + [-11664m^4 +27216m³ -324m² -48384m +37440] = 0Wait, the discriminant D is equal to:(108m² -126m)^2 -4AC = [11664m^4 -27216m^3 +15876m²] + [-11664m^4 +27216m³ -324m² -48384m +37440]So combine term by term:11664m^4 -11664m^4 = 0-27216m^3 +27216m^3 = 015876m² -324m² = 15552m²-48384m remains+37440 remainsSo D = 15552m² -48384m +37440Set discriminant D = 0:15552m² -48384m +37440 = 0Let me simplify this equation. First, notice that all coefficients are divisible by 48:15552 ÷ 48 = 32448384 ÷ 48 = 100837440 ÷ 48 = 780So divide each term by 48:324m² -1008m +780 = 0Still, check if we can divide further. Let's see:324, 1008, 780: GCD is 12? 324 ÷12=27, 1008 ÷12=84, 780 ÷12=65. Yes.Divide by 12:27m² -84m +65 = 0So the quadratic equation in m is 27m² -84m +65 = 0Now, let's solve this quadratic equation. Let's compute the discriminant:D' = (-84)^2 -4*27*65 = 7056 - 4*27*65Compute 4*27 = 108, 108*65 = 7020Thus, D' = 7056 -7020 =36Square root of 36 is 6. So solutions are:m = [84 ±6]/(2*27) = (84 ±6)/54Thus:m = (84 +6)/54 =90/54=5/3 ≈1.6667m = (84 -6)/54=78/54=13/9 ≈1.4444So the slopes are m =5/3 and m=13/9.Therefore, the equations of the tangent lines are:For m=5/3:c =7 -6*(5/3)=7 -10= -3So equation: y = (5/3)x -3Multiply both sides by 3: 3y =5x -9 =>5x -3y -9=0For m=13/9:c=7 -6*(13/9)=7 -26/3= (21/3 -26/3)= -5/3So equation: y=(13/9)x -5/3Multiply both sides by 9: 9y=13x -15 =>13x -9y -15=0Therefore, there are two tangent lines: 5x -3y -9=0 and13x -9y -15=0.But let me verify these results to make sure.First, check if the lines pass through (6,7):For 5x -3y -9=0:Left-hand side:5*6 -3*7 -9=30 -21 -9=0, which is correct.For13x -9y -15=0:13*6 -9*7 -15=78 -63 -15=0, correct.Now, check if they are tangent to the hyperbola.Take the first line:5x -3y =9. Let's substitute into the hyperbola.Express y from the line:3y=5x -9 => y=(5x -9)/3Plug into hyperbola equation:x²/9 - [(5x -9)/3]^2 /16 =1Compute [(5x -9)/3]^2 = (25x² -90x +81)/9Divide by16: (25x² -90x +81)/(9*16)= (25x² -90x +81)/144So equation becomes:x²/9 - (25x² -90x +81)/144 =1Multiply both sides by144:16x² - (25x² -90x +81)=14416x² -25x² +90x -81=144-9x² +90x -81 -144=0-9x² +90x -225=0Divide by -9:x² -10x +25=0This factors as (x -5)^2=0, which has a repeated root at x=5. So yes, it's tangent at x=5. Then y=(5*5 -9)/3=(25 -9)/3=16/3≈5.333. So the point of tangency is (5,16/3). Let's check if this is on the hyperbola:x²/9 - y²/16=25/9 - (256/9)/16=25/9 -16/9=9/9=1. Correct.Similarly, check the second line:13x -9y=15 => y=(13x -15)/9Plug into hyperbola:x²/9 - [(13x -15)/9]^2 /16=1Compute [(13x -15)/9]^2=(169x² -390x +225)/81Divide by16: (169x² -390x +225)/(81*16)= (169x² -390x +225)/1296So equation becomes:x²/9 - (169x² -390x +225)/1296 =1Multiply both sides by1296:144x² - (169x² -390x +225)=1296144x² -169x² +390x -225=1296-25x² +390x -225 -1296=0-25x² +390x -1521=0Multiply by -1:25x² -390x +1521=0Compute discriminant:D=(-390)^2 -4*25*1521=152100 -4*25*1521Compute 4*25=100, so 100*1521=152100Thus, D=152100 -152100=0. So repeated root. Therefore, tangent.Solve for x:x=(390)/(2*25)=390/50=7.8=39/5=7.8Then y=(13*(39/5)-15)/9=(507/5 -75/5)/9=(432/5)/9=48/5=9.6Check if (39/5,48/5) is on the hyperbola:x²/9 - y²/16=(1521/25)/9 - (2304/25)/16=1521/(225) -2304/(400)=Simplify:1521/225 = 6.762304/400=5.76So 6.76 -5.76=1, which is correct. So yes, that point is on the hyperbola.Therefore, both lines are indeed tangent to the hyperbola and pass through (6,7).So the equations are 5x -3y -9=0 and13x -9y -15=0.Alternatively, written in standard form:5x - 3y = 9 and 13x - 9y =15.Alternatively, if they want them in slope-intercept form, but the problem just says "the equation", so either form is acceptable, but likely standard form.Final AnswerThe equations of the tangent lines are boxed{5x - 3y = 9} and boxed{13x - 9y = 15}.</think>To find the equation of the line passing through the point ( P(6, 7) ) and tangent to the hyperbola (frac{x^2}{9} - frac{y^2}{16} = 1), we start by considering the general form of a line passing through ( P(6, 7) ), which is ( y = mx + c ). Since the line passes through ( P(6, 7) ), we have ( 7 = 6m + c ), giving ( c = 7 - 6m ). Thus, the equation of the line is ( y = mx + 7 - 6m ).Substituting this into the hyperbola equation:[frac{x^2}{9} - frac{(mx + 7 - 6m)^2}{16} = 1]Expanding and simplifying this equation, we get a quadratic equation in ( x ):[(16 - 9m^2)x^2 + (108m^2 - 126m)x + (-585 + 756m - 324m^2) = 0]For the line to be tangent to the hyperbola, the discriminant of this quadratic equation must be zero. Calculating the discriminant and setting it to zero, we solve for ( m ):[15552m^2 - 48384m + 37440 = 0]Simplifying this equation, we find:[27m^2 - 84m + 65 = 0]Solving this quadratic equation, we get the slopes ( m = frac{5}{3} ) and ( m = frac{13}{9} ). Using these slopes, we find the corresponding intercepts ( c ):- For ( m = frac{5}{3} ), ( c = -3 ), giving the line equation ( y = frac{5}{3}x - 3 ), which simplifies to ( 5x - 3y = 9 ).- For ( m = frac{13}{9} ), ( c = -frac{5}{3} ), giving the line equation ( y = frac{13}{9}x - frac{5}{3} ), which simplifies to ( 13x - 9y = 15 ).Both lines pass through ( P(6, 7) ) and are tangent to the hyperbola. Verifying these points confirms their validity. Final AnswerThe equations of the tangent lines are boxed{5x - 3y = 9} and boxed{13x - 9y = 15}.]

🔑:Solving by using property 5, combining the equations 6 A+7 B+C=0 and 9 A^{2}-16 B^{2}=C^{2} to eliminate C, we get(3 A+5 B)(9 A+13 B)=0 text {. }Solving for A=-frac{5}{3} B or A=-frac{13}{9} B, thus we get C=3 B or C=frac{5}{3} B, meaning the required tangent equations are -frac{5}{3} B x+B y+3 B=0 and -frac{13}{9} B x+B y+frac{5}{3} B=0, which are 5 x-3 y-9=0 and 13 x-9 y-15=0 respectively.

🧠:[<think>Okay, let me try to figure out this problem. So, the question is about predicting Zhang Honghong's weight based on his height using a given regression equation. Hmm, let me start by understanding the details.First, the regression equation is provided as y = 0.72x - 58.5. Here, y represents weight in kilograms, and x is height in centimeters. Zhang Honghong's height is given as 1.78 meters. Since the equation uses centimeters, I need to convert his height from meters to centimeters. Right, 1 meter is 100 centimeters, so 1.78 meters would be 1.78 * 100. Let me calculate that. 1.78 times 100 is 178 centimeters. So, x is 178 cm.Now, plugging this into the regression equation. The equation is y = 0.72x - 58.5. Substituting x with 178, it becomes y = 0.72 * 178 - 58.5. Let me compute 0.72 multiplied by 178 first.Breaking it down, 0.72 times 100 is 72, so 0.72 times 100 is 72. Then 0.72 times 70 is 50.4 (since 0.72*70=50.4), and 0.72 times 8 is 5.76. Adding those together: 72 + 50.4 + 5.76. Let's add 72 and 50.4 first: that's 122.4. Then add 5.76 to get 128.16. So, 0.72 * 178 = 128.16.Now subtract 58.5 from that result. So, 128.16 - 58.5. Let me do that subtraction. 128.16 minus 50 is 78.16, then minus 8.5 more is 69.66. So, y equals 69.66 kilograms.Wait, but the problem mentions that Zhang Honghong is "neither fat nor thin," which might mean that the regression equation is appropriate here because it's under normal circumstances. So, the answer should be around 69.66 kg. Rounding that to a reasonable number, maybe one decimal place or a whole number. Since weight is often expressed in whole numbers, perhaps 70 kg? But let me check my calculations again to be sure.Double-checking the multiplication: 178 * 0.72. Let's do it another way. 178 * 0.7 = 124.6, and 178 * 0.02 = 3.56. Adding those gives 124.6 + 3.56 = 128.16. Yes, that's correct. Then 128.16 - 58.5 is indeed 69.66. So, 69.66 kg. But maybe the answer expects rounding to the nearest whole number. So, 69.66 is approximately 70 kg. However, sometimes in these contexts, they might keep one decimal place. The problem says "around," so either might be acceptable, but in many cases, weight is given as a whole number. Let me see if the original equation has any decimal precision. The coefficients are given to two decimal places, but the final answer could be rounded to a whole number. Alternatively, maybe 69.7 kg if keeping one decimal. But the problem is in Chinese, and sometimes in such problems, they might expect a whole number. Let me check again. Alternatively, maybe there is a mistake in my calculation. Let me verify once more. 0.72 multiplied by 178. Let's do 178 times 0.7: 178 * 0.7 = 124.6. 178 times 0.02 is 3.56. So 124.6 + 3.56 is indeed 128.16. Then subtract 58.5: 128.16 - 58.5. Let me subtract step by step. 128.16 - 50 = 78.16, then subtract 8.5 more: 78.16 - 8.5 = 69.66. So that's correct. Alternatively, maybe I can calculate 178 - 58.5 / 0.72? Wait no, the equation is linear regression, so it's y = 0.72x -58.5. So, the calculation is correct as is. Therefore, 69.66 kg. Since the question says "around," perhaps 70 kg is acceptable. However, depending on the context, sometimes 0.66 is closer to 0.7, so 69.7 kg. But in China, they might use one decimal place for such measurements. Alternatively, maybe the answer is presented as 69.7. Let me check if 0.72*178 is indeed correct.Alternatively, 178 divided by 100 is 1.78, so 0.72*178. Let me compute 72*178 and then divide by 100. Wait, 72*178. Let me compute 70*178 + 2*178 = 12,460 + 356 = 12,816. Then divide by 100: 128.16. So that's correct. Then subtract 58.5 gives 69.66. So, that's precise.But since weight is often a decimal in kilograms, like people can be 69.6 kg. However, in the problem statement, the regression equation is given with two decimal places, but maybe the answer is expected to be an integer. If we round 69.66 to the nearest whole number, it's 70. However, depending on the convention, sometimes 0.5 and above rounds up, so 69.66 is closer to 70 than 69. So, 70 kg. But maybe the problem expects the exact value. Let me check the question again.The problem says "His weight should be around ______ kg." The word "around" suggests that an approximate value is acceptable, so either 69.7 or 70. However, given that the original equation uses two decimal places, but the answer is likely expected to be a whole number. In China, typically, weight is often stated as a whole number in such contexts. For example, if someone is 69.6 kg, they might say 70 kg. So, I think 70 is the answer they expect here. But to be precise, the exact value is 69.66, which is approximately 69.7 kg. But maybe the problem expects rounding to the nearest whole number. Alternatively, let's consider that 0.72 times 178 is 128.16, minus 58.5 is 69.66. So, 69.66 kg. If we need to give an answer "around" that, maybe 70 kg. However, another way is to check if 1.78 meters is exactly 178 cm. Since 1.78 meters is 178.0 cm, no issues there. So, the calculation is precise. Wait, but the problem says "Zhang Honghong, who is neither fat nor thin," which might mean that his weight is exactly as predicted by the regression equation, so we don't need to adjust for any residuals. So, the answer should be the exact calculation, which is 69.66 kg. However, in the context of the problem, maybe it's intended to round to the nearest whole number. Let me see if 0.66 is closer to 0.7, so 69.7, which would be approximately 70 when rounded to the nearest whole number. But different rounding conventions might apply. Alternatively, perhaps the problem expects the answer to be given as a whole number, so 70 kg. Since 0.66 is two-thirds, which is more than half, so rounding up. Therefore, 70 kg. However, in some educational contexts, they might prefer to keep one decimal. But since the original equation has two decimal places, but the answer is about weight, which is commonly given to one decimal or as a whole number. Wait, let me check the units. The equation is in kilograms, and the answer should be in kg. In many cases, people's weight is reported in whole kilograms unless precision is required. Since Zhang is "neither fat nor thin," the exact decimal might not be necessary, so 70 kg is reasonable. Alternatively, the problem might accept 69.7, but perhaps the answer is intended to be an integer. Let me think. Alternatively, maybe there was a miscalculation. Wait, 178 cm multiplied by 0.72. Let me verify again. 178 * 0.7: 178*0.7 = 124.6. 178*0.02 = 3.56. So 124.6 + 3.56 = 128.16. Correct. 128.16 - 58.5: 128.16 - 58.16 = 70, then +0.34 (because 58.5 is 58.16 + 0.34), so 70 - 0.34 = 69.66. Correct. So, the exact value is 69.66 kg, which is approximately 70 kg when rounded. Alternatively, maybe the answer is 70 kg. But given that 0.66 is two-thirds, maybe the problem expects rounding to the nearest whole number. So, 70 kg. Alternatively, in some contexts, they might keep one decimal, so 69.7 kg. However, in China, when people talk about weight, they often use whole numbers. For example, someone might say they weigh 70 kilograms rather than 69.6. Therefore, considering this cultural context, the answer is likely 70 kg. But to be thorough, let me check if there's any other possible mistake. For instance, converting meters to centimeters. 1.78 meters is indeed 178 centimeters. Yes, that's correct. The regression equation is for people aged 18 to 38, which presumably includes Zhang Honghong, as no age is given, but since it's stated under normal circumstances, and he's neither fat nor thin, the equation applies. So, the calculation seems correct. The answer is 69.66 kg. If we need to present it as around, then 70 kg is appropriate. However, since the problem might expect an exact answer based on the calculation, even with decimals, but in the format of the answer, they want it boxed. Let me check the instructions again. The user says "put your final answer within boxed{}". Typically, in such mathematical problems, if the calculation results in a decimal, they might expect it to be rounded to a reasonable figure. But given the original equation has two decimal places, but the answer here is 69.66, which could be written as 69.7 if rounding to one decimal. Alternatively, since 0.66 is two decimal places, but maybe they expect two decimal places here. Wait, the problem says "around," so maybe it's okay to present 70 kg. Let me think again. If someone's height is 178 cm, and the equation is y = 0.72x -58.5, then plugging in, the exact value is 69.66 kg. If we round to the nearest whole number, that's 70 kg. Since the problem is in Chinese educational context, maybe they expect the whole number. But to be safe, perhaps write both? Wait, no, the instructions say to put the final answer within a box. So probably, they expect the exact value, 69.66, but maybe rounded. However, given that weight is usually given as a whole number or one decimal, perhaps 69.7 or 70. Wait, let me check a sample. If someone's height is 170 cm, then using the equation: 0.72*170 = 122.4, minus 58.5 is 63.9 kg. So, approximately 64 kg. Similarly, 178 cm gives 69.66, so 70 kg. So maybe the answer is 70 kg. Alternatively, maybe the problem expects us to keep one decimal place. Let me see: 0.72 is two decimal places, 58.5 is one decimal place. The result is 69.66, which is two decimal places. If we follow significant figures, the coefficients have three significant figures (0.72) and three as well (58.5). The height is given as 1.78 meters, which is three significant figures. Therefore, the result should have three significant figures. 69.66 has four significant figures. So, rounding to three, it would be 69.7 kg. But perhaps in the context of the problem, since it's a weight prediction, they might just round to the nearest whole number for simplicity. But given that 69.66 is 69 and two-thirds, which is closer to 70 than 69, maybe it's better to round up. Alternatively, maybe the problem expects the answer without rounding, just the exact calculation. But the question says "around," so rounding is acceptable. Hmm. Given this is a regression equation, the output is an estimate, so it's appropriate to present it as a rounded number. Since 69.66 is approximately 70, and given the usual conventions, 70 kg is a reasonable answer. Therefore, I think the answer should be 70 kg.Final Answerboxed{70}</think>Zhang Honghong's height is given as 1.78 meters, which converts to 178 centimeters. Using the regression equation ( y = 0.72x - 58.5 ), where ( y ) is weight in kilograms and ( x ) is height in centimeters, we substitute ( x = 178 ):[y = 0.72 times 178 - 58.5]First, calculate ( 0.72 times 178 ):[0.72 times 178 = 128.16]Next, subtract 58.5 from the result:[128.16 - 58.5 = 69.66]Since the problem states that Zhang Honghong's weight should be "around" a certain value, we round 69.66 to the nearest whole number, which is 70. Therefore, the answer is:[boxed{70}],<think>Okay, so I need to figure out Zhang Honghong's weight based on the given regression equation. Let me start by understanding what the problem is asking.First, the regression equation is provided as y = 0.72x - 58.5, where y is weight in kilograms and x is height in centimeters. Zhang Honghong's height is given as 1.78 meters. But wait, the equation uses centimeters, so I need to convert his height from meters to centimeters. Right, 1 meter is 100 centimeters, so 1.78 meters is 1.78 * 100. Let me calculate that. 1.78 * 100 = 178 centimeters. Okay, so x = 178 cm.Now, plug that into the regression equation. The equation is y = 0.72x - 58.5. Substituting x with 178, it becomes y = 0.72 * 178 - 58.5. Let me compute 0.72 multiplied by 178 first.Hmm, let me break that down. 0.72 is the same as 72/100. So, 72/100 * 178. Maybe I can compute 178 * 70 first and then 178 * 2, then add them together and divide by 100?Wait, 70% of 178 is 0.7 * 178. Let's calculate 0.7 * 178. 178 * 0.7: 100*0.7=70, 70*0.7=49, 8*0.7=5.6. Adding those up: 70 + 49 + 5.6 = 124.6. Then 2% of 178 is 0.02*178 = 3.56. So adding 70% and 2% gives 124.6 + 3.56 = 128.16. So 0.72 * 178 is 128.16.Alternatively, maybe a quicker way: 178 * 0.72. Let me check that again. 178 * 0.7 is 124.6, 178 * 0.02 is 3.56, so total is 124.6 + 3.56 = 128.16. Yes, that's correct.So then, y = 128.16 - 58.5. Let me subtract that. 128.16 minus 58.5. 128.16 - 50 = 78.1678.16 - 8.5 = 69.66So, 128.16 - 58.5 = 69.66 kg.Therefore, Zhang Honghong's predicted weight is approximately 69.66 kilograms. Since the problem says he's neither fat nor thin, this should be around the expected weight. But maybe we need to round it to a reasonable number? The question says "around" so probably rounding to the nearest whole number or one decimal place.69.66 kg is approximately 69.7 kg, or 70 kg if rounding to the nearest whole number. Considering typical weight discussions, people often use whole numbers, so 70 kg seems reasonable. Let me check the calculation again to make sure I didn't make any errors.Height: 1.78 m = 178 cm. Correct.Regression equation: y = 0.72x - 58.5. Plug in x=178.0.72*178: 178*0.7=124.6; 178*0.02=3.56. 124.6+3.56=128.16. Then subtract 58.5: 128.16-58.5=69.66. Yep, that's correct.So 69.66 kg. Depending on the context, maybe keep one decimal place: 69.7 kg. But in China, do they commonly use decimal places for weight? Maybe just round to the nearest whole number, so 70 kg. Since the question asks for "around," either 69.7 or 70 is acceptable, but likely expecting 70 kg.Wait, let me verify once more. If I compute 0.72*178, perhaps another way. Let's do 178*0.72:Breakdown 178 * 0.72 = 178*(70 + 2)/100 = (178*70 + 178*2)/100.178*70: 178*7=1246, so 1246*10=12460.178*2=356.Sum: 12460 + 356 = 12816.Divide by 100: 12816/100 = 128.16. Correct. Then subtract 58.5: 128.16 - 58.5 = 69.66. Correct. So no mistake in calculation.Therefore, the answer should be approximately 70 kg when rounded to the nearest whole number. That's reasonable for someone who is 1.78 meters tall and described as neither fat nor thin. For example, a BMI check: BMI = weight (kg) / (height (m))^2. Let's calculate that with 69.66 kg and 1.78 m.BMI = 69.66 / (1.78)^2. Compute 1.78 squared: 1.78*1.78. Let's calculate that:1.78 * 1.78: 1*1=1, 1*0.78=0.78, 0.78*1=0.78, 0.78*0.78=0.6084.Wait, no, that's not the right way. Let me do it properly.(1.78)^2 = (1 + 0.78)^2 = 1^2 + 2*1*0.78 + (0.78)^2 = 1 + 1.56 + 0.6084 = 1 + 1.56 = 2.56; 2.56 + 0.6084 = 3.1684.So BMI = 69.66 / 3.1684 ≈ 22.0. A BMI of 22 is considered normal, which aligns with "neither fat nor thin." So that makes sense. Therefore, the answer is approximately 70 kg, which gives a BMI of 22, perfectly normal. So that seems correct.Hence, the answer should be 70 kg.Final AnswerZhang Honghong's weight should be around boxed{70} kg.</think>Zhang Honghong's height is given as 1.78 meters, which converts to 178 centimeters. Using the regression equation ( y = 0.72x - 58.5 ), where ( y ) is weight in kilograms and ( x ) is height in centimeters, we substitute ( x = 178 ):[y = 0.72 times 178 - 58.5]First, calculate ( 0.72 times 178 ):[0.72 times 178 = 128.16]Next, subtract 58.5 from the result:[128.16 - 58.5 = 69.66]Rounding 69.66 to the nearest whole number gives 70. A BMI check confirms this is a normal weight for someone of 1.78 meters height, resulting in a BMI of approximately 22.[boxed{70}]]

🔑:Given the problem, we have the equation y=0.72 times 178 - 58.5 approx 70 (kg).Therefore, the answer is boxed{70}.By substituting x=178 into the regression equation for weight y (kg) based on height x (cm), we can estimate the weight. This question tests the regression equation and examines the students' calculation abilities, which is quite basic.