🧠:[<think>Okay, so I need to find the conjugate of the complex number z, given that (3 - 4i)z = 1 + 2i. The options are A through D, each with different fractions. Hmm, let's break this down step by step.First, I remember that to solve for z in a complex equation like this, I should divide both sides by (3 - 4i). That way, z will be isolated. So, z = (1 + 2i)/(3 - 4i). But dividing complex numbers can be tricky. Oh, right! The standard method is to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (3 - 4i) is (3 + 4i). Let me write that down.So, z = [(1 + 2i)(3 + 4i)] / [(3 - 4i)(3 + 4i)]. Multiplying those out should simplify the denominator into a real number, which makes it easier to handle.Starting with the denominator: (3 - 4i)(3 + 4i). Using the difference of squares formula, (a - b)(a + b) = a² - b². Here, a is 3 and b is 4i. So, 3² - (4i)². That's 9 - 16i². Since i² is -1, this becomes 9 - 16(-1) = 9 + 16 = 25. So the denominator simplifies to 25. That's straightforward.Now the numerator: (1 + 2i)(3 + 4i). Let's use the distributive property (FOIL) here. First, multiply 1*3 = 3. Then, 1*4i = 4i. Next, 2i*3 = 6i. Finally, 2i*4i = 8i². Adding all these together: 3 + 4i + 6i + 8i². Combine like terms: 3 + (4i + 6i) + 8i² = 3 + 10i + 8i². Again, since i² = -1, 8i² = -8. So, 3 - 8 + 10i = -5 + 10i. Therefore, the numerator is -5 + 10i.Putting it all together, z = (-5 + 10i)/25. Let's split this into real and imaginary parts: (-5/25) + (10i)/25. Simplifying the fractions: -1/5 + (2/5)i. So, z = -1/5 + (2/5)i.But the question asks for the conjugate of z. The conjugate of a complex number a + bi is a - bi. So, if z is -1/5 + (2/5)i, then its conjugate should be -1/5 - (2/5)i. Looking at the options, that's option C. Wait, but let me check the options again.Option C is -1/5 - 2/5i, which matches my calculation. But let me double-check my steps to make sure I didn't make a mistake.Starting with (3 - 4i)z = 1 + 2i. Solving for z: z = (1 + 2i)/(3 - 4i). Multiply numerator and denominator by (3 + 4i):Numerator: (1 + 2i)(3 + 4i). Let me compute that again. 1*3=3, 1*4i=4i, 2i*3=6i, 2i*4i=8i². So, 3 + 4i + 6i + 8i². Combine terms: 3 + 10i + 8(-1) = 3 + 10i -8 = -5 + 10i. That's correct.Denominator: (3 - 4i)(3 + 4i) = 9 + 12i -12i -16i² = 9 - (-16) = 25. Correct.So z is (-5 + 10i)/25 = (-1/5) + (2/5)i. Therefore, conjugate of z is (-1/5) - (2/5)i. Which is option C. But wait, looking back at the options given:A: -1/5 + 2/5iB: 1/5 + 2/5iC: -1/5 - 2/5iD: 1/5 - 2/5iSo yes, option C is the conjugate. But let me check again because sometimes when I do these problems, I might mix up signs.Wait, let me verify the conjugate again. If z = a + bi, then conjugate is a - bi. So here, a is -1/5 and b is 2/5. Therefore, conjugate is -1/5 - 2/5i. Which is exactly option C. So that should be the correct answer.But wait, another way to check: maybe if I compute z conjugate first. Let's see, if (3 - 4i)z = 1 + 2i, then taking conjugates of both sides, the equation becomes (3 + 4i)z_conjugate = 1 - 2i. Therefore, z_conjugate = (1 - 2i)/(3 + 4i). Then, to compute this, multiply numerator and denominator by the conjugate of (3 + 4i), which is (3 - 4i). So:z_conjugate = (1 - 2i)(3 - 4i) / (3 + 4i)(3 - 4i)Compute denominator first: same as before, 9 - (4i)^2 = 9 - (-16) = 25.Numerator: (1 - 2i)(3 - 4i). Multiply out: 1*3 = 3, 1*(-4i) = -4i, (-2i)*3 = -6i, (-2i)*(-4i) = 8i². So total is 3 -4i -6i +8i². Combine terms: 3 -10i +8(-1) = 3 -10i -8 = -5 -10i. Therefore, z_conjugate = (-5 -10i)/25 = (-1/5) - (2/5)i. Which is again option C. So that confirms it.But wait, just to make sure, let me check if the initial answer might have a mistake. For instance, maybe the question was to compute z conjugate and I thought it was z, but no, the problem says "the conjugate of z is", so yes, it's correct. Therefore, the answer is C.But looking at the options, I notice that option D is 1/5 - 2/5i. Could there be a miscalculation? Let's see. Let me try the alternative method again:Original equation: (3 - 4i)z = 1 + 2i.If I solve for z by multiplying both sides by the conjugate of (3 -4i), but actually, we need to divide both sides by (3 -4i). So, z = (1 + 2i)/(3 -4i). Multiply numerator and denominator by (3 +4i):Numerator: (1 + 2i)(3 +4i) = 1*3 +1*4i +2i*3 +2i*4i = 3 +4i +6i +8i² = 3 +10i -8 = -5 +10i.Denominator: 25, as before. So z = (-5 +10i)/25 = -1/5 + 2/5i. Then conjugate is -1/5 -2/5i. Yes, that's correct. So option C.Alternatively, perhaps I miscalculated when using the alternative method with conjugating both sides. Let's check that again.Original equation: (3 -4i) z = 1 +2i.Taking conjugates: (3 +4i) z_conj = 1 -2i.Then, solving for z_conj: z_conj = (1 -2i)/(3 +4i).Multiply numerator and denominator by (3 -4i):Numerator: (1 -2i)(3 -4i) = 1*3 +1*(-4i) + (-2i)*3 + (-2i)*(-4i) = 3 -4i -6i +8i² = 3 -10i -8 = -5 -10i.Denominator: 25. So z_conj = (-5 -10i)/25 = -1/5 -2/5i. Same result. So yes, option C is correct.But looking at the answer options, the letters are A, B, C, D, with C being -1/5 -2/5i. So, the answer is C. Therefore, the correct choice is C. Wait, but wait, let me check once more.Alternatively, if I miscalculated the numerator when multiplying (1 +2i)(3 +4i). Let me check that step again.1*3 = 3, 1*4i=4i, 2i*3=6i, 2i*4i=8i². So 3 +4i +6i +8i². Then 3 +10i +8*(-1)=3 +10i -8= -5 +10i. Correct.Denominator: 3*3=9, 3*4i=12i, -4i*3=-12i, -4i*4i=-16i². So 9 +12i -12i -16i²=9 -(-16)=25. Correct.So z is (-5 +10i)/25 = -1/5 + 2/5i. So conjugate is -1/5 -2/5i. So option C. Yes, seems right.Alternatively, perhaps the problem is designed to test whether I confuse z with its conjugate. Maybe some of the answer options are z itself. For example, option A is -1/5 +2/5i, which is z. So if someone thought the question was asking for z, they might choose A, but the question asks for the conjugate. So C is correct.Therefore, after multiple checks, I'm confident the answer is C.Final Answerboxed{C}</think>Given that (i) is the imaginary unit, we need to find the conjugate of the complex number (z) that satisfies ((3 - 4i)z = 1 + 2i).First, solve for (z) by dividing both sides by (3 - 4i):[z = frac{1 + 2i}{3 - 4i}]To simplify this, multiply the numerator and the denominator by the conjugate of the denominator, (3 + 4i):[z = frac{(1 + 2i)(3 + 4i)}{(3 - 4i)(3 + 4i)}]Calculate the denominator using the difference of squares formula:[(3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25]Expand the numerator using the distributive property (FOIL):[(1 + 2i)(3 + 4i) = 1 cdot 3 + 1 cdot 4i + 2i cdot 3 + 2i cdot 4i = 3 + 4i + 6i + 8i^2]Combine like terms and substitute (i^2 = -1):[3 + 4i + 6i + 8(-1) = 3 + 10i - 8 = -5 + 10i]Thus, the simplified form of (z) is:[z = frac{-5 + 10i}{25} = frac{-1}{5} + frac{2}{5}i]The conjugate of (z = -frac{1}{5} + frac{2}{5}i) is:[overline{z} = -frac{1}{5} - frac{2}{5}i]Therefore, the conjugate of (z) is (boxed{C}).,<think>Okay, let me try to figure this problem out. So, we have a complex number z, and it's given that (3 - 4i)z = 1 + 2i. We need to find the conjugate of z, and then choose the correct option from A to D. First, I remember that to solve for z in such equations, we usually divide both sides by the complex number (3 - 4i). But dividing by a complex number can be a bit tricky. I think the standard method is to multiply the numerator and denominator by the conjugate of the denominator to rationalize it. So, the conjugate of (3 - 4i) is (3 + 4i). Let me try that.So starting with the equation:(3 - 4i)z = 1 + 2iTo solve for z, divide both sides by (3 - 4i):z = (1 + 2i) / (3 - 4i)Now, multiply numerator and denominator by the conjugate of the denominator, which is (3 + 4i):z = [(1 + 2i)(3 + 4i)] / [(3 - 4i)(3 + 4i)]Let me compute both numerator and denominator separately. Starting with the denominator:(3 - 4i)(3 + 4i) = 3*3 + 3*4i - 4i*3 - 4i*4iCalculate each term:3*3 = 93*4i = 12i-4i*3 = -12i-4i*4i = -16i²Now, combine like terms. The 12i and -12i cancel each other out. And since i² = -1, -16i² becomes +16. So denominator = 9 + 0i +16 = 25. So the denominator simplifies to 25. That's straightforward.Now the numerator:(1 + 2i)(3 + 4i) = 1*3 + 1*4i + 2i*3 + 2i*4iCalculate each term:1*3 = 31*4i = 4i2i*3 = 6i2i*4i = 8i²Combine like terms:Real parts: 3 and 8i². Since i² = -1, 8i² = -8. So real parts: 3 -8 = -5.Imaginary parts: 4i + 6i = 10i.So numerator = -5 + 10i.Therefore, z = (-5 + 10i)/25 = (-5/25) + (10i)/25 = -1/5 + (2/5)i.So z = -1/5 + (2/5)i. But the question is asking for the conjugate of z. The conjugate of a complex number a + bi is a - bi. So, taking the conjugate of z:Conjugate(z) = -1/5 - (2/5)i.Looking at the options, that's option C: -1/5 - 2/5i. Wait, but let me check the options again.Wait, let me verify the options:A: -1/5 + 2/5iB: 1/5 + 2/5iC: -1/5 - 2/5iD: 1/5 - 2/5iSo according to my calculation, conjugate(z) is -1/5 - 2/5i, which is option C. But hold on, let me check again because sometimes I might have made a mistake in the sign.Wait, when I calculated z, I got z = -1/5 + 2/5i. Then the conjugate should replace the imaginary part with its negative. So, yes, that would be -1/5 - 2/5i, which is option C. Hmm. But let me cross-verify my calculation because sometimes when I do these problems quickly, I might have messed up a sign.Let me redo the numerator just to confirm:(1 + 2i)(3 + 4i) = 1*3 + 1*4i + 2i*3 + 2i*4i= 3 + 4i + 6i + 8i²= 3 + 10i + 8(-1)= 3 + 10i - 8= (3 - 8) + 10i= -5 + 10i. That's correct.Denominator is 25. So z = (-5 + 10i)/25 = (-1 + 2i)/5 = -1/5 + 2/5i. Then conjugate is -1/5 - 2/5i. That's option C. But wait, the answer options might have a different order? Let me check again.Yes, options:A: -1/5 + 2/5iB: 1/5 + 2/5iC: -1/5 - 2/5iD: 1/5 - 2/5iSo the conjugate is -1/5 - 2/5i, which is C. Hmm. Wait, but let me see if the problem is written correctly. The original equation is (3 - 4i)z = 1 + 2i, right?Yes. So solving for z, as above, gives z = (-1 + 2i)/5. Then conjugate is (-1 -2i)/5, which is -1/5 -2/5i. So answer is C.But wait, hold on. Let me check if I didn't make any miscalculations here because sometimes even small mistakes can happen. Let me verify with an alternative approach.Alternatively, suppose we let z = a + bi, then compute (3 -4i)(a + bi) and set it equal to 1 + 2i. Then solve for a and b.So expanding (3 -4i)(a + bi):= 3a + 3bi -4i a -4i*bi= 3a + 3bi -4ai -4b i²Simplify:= 3a + (3b -4a)i + 4b (since i² = -1)Combine real and imaginary parts:Real: 3a + 4bImaginary: (3b -4a)iSet equal to 1 + 2i:So,3a + 4b = 13b -4a = 2So we have a system of equations:1) 3a + 4b = 12) -4a + 3b = 2Let me solve this system. Let's write them again:3a + 4b = 1 ...(1)-4a +3b = 2 ...(2)Let me use the elimination method. Multiply equation (1) by 4 and equation (2) by 3 to eliminate a:Equation (1) *4: 12a +16b =4Equation (2)*3: -12a +9b =6Now add them:(12a -12a) + (16b +9b) =4 +60a +25b =10Thus, 25b=10 => b=10/25=2/5.Then substitute back into equation (1):3a +4*(2/5)=13a +8/5=13a=1 -8/5= (5/5 -8/5)= -3/5Thus, a= (-3/5)/3= -1/5.Therefore, z =a +bi= -1/5 + (2/5)i. Then the conjugate of z is -1/5 -2/5i, which is option C. So same result as before. Therefore, answer is C.But wait, the answer options don't have C as the last one. Let me check again.Wait the options are:A: -1/5 + 2/5iB: 1/5 + 2/5iC: -1/5 - 2/5iD:1/5 - 2/5iSo yes, option C is -1/5 -2/5i, which is the conjugate. Therefore, the answer should be C.But wait, just to be 100% sure, let me check once again.Alternatively, suppose I use the formula for the conjugate. If z = (1 +2i)/(3 -4i), then conjugate(z) = conjugate(1 +2i)/conjugate(3 -4i) = (1 -2i)/(3 +4i). Then compute that.Let me compute (1 -2i)/(3 +4i). Multiply numerator and denominator by (3 -4i):[(1 -2i)(3 -4i)] / [(3 +4i)(3 -4i)]Denominator is same as before: 9 +16=25.Numerator:(1)(3) +1*(-4i) -2i*3 + (-2i)*(-4i)=3 -4i -6i +8i²=3 -10i +8(-1)=3 -10i -8= -5 -10iThus, conjugate(z)= (-5 -10i)/25 = (-5/25) -10i/25 = -1/5 -2/5i, same result. Therefore, option C is correct. So answer is C.But wait, the answer options given here... I need to check if the user didn't make a typo. But according to the problem statement, the options are:A: -1/5 + 2/5iB:1/5 + 2/5iC: -1/5 -2/5iD:1/5 -2/5iTherefore, yes, the answer is C. But in the original problem, in the options, the conjugate is C. So why do I have a doubt? Wait, perhaps I misread the problem. Let me check once again.The problem says: If the complex number z satisfies (3 -4i)z =1 +2i, then the conjugate of z is which option. So yes, according to all methods, the answer is C. So I think it's correct.But wait, hold on. When I first solved for z, I got z = -1/5 + 2/5i, so conjugate is -1/5 -2/5i, which is option C. But looking at the options, maybe there's a trick here? For example, sometimes people get confused between z and conjugate(z). Let me check if the problem is asking for conjugate(z) or z. The problem clearly states "the conjugate of z", so that's correct. So yes, answer is C.But let me check online or with another method. Wait, perhaps cross-verifying with another approach.Alternatively, compute z and conjugate(z) using matrix representation or something else, but that might complicate. Alternatively, use the property that conjugate of (az) = conjugate(a) * conjugate(z). But maybe that's not helpful here.Alternatively, suppose we take conjugate of both sides of the original equation. So:Conjugate[(3 -4i)z] = Conjugate[1 +2i]Which is Conjugate(3 -4i)*Conjugate(z) = 1 -2iThus, (3 +4i)*Conjugate(z) =1 -2iTherefore, solving for Conjugate(z):Conjugate(z)= (1 -2i)/(3 +4i)Then compute this. Multiply numerator and denominator by (3 -4i):[(1 -2i)(3 -4i)] / [(3 +4i)(3 -4i)]Denominator is 25 again.Numerator:1*3 +1*(-4i) -2i*3 + (-2i)*(-4i)=3 -4i -6i +8i²=3 -10i -8= -5 -10iSo Conjugate(z)= (-5 -10i)/25= -1/5 -2/5i. Which is same as before. So this confirms that the answer is C.Therefore, the correct answer is option C. So the answer is C.But wait, hold on. Let me check once more if the calculation is correct because sometimes even after multiple checks, a mistake can slip through.Original equation: (3 -4i)z =1 +2i. Solve for z.z = (1 +2i)/(3 -4i). Multiply numerator and denominator by (3 +4i):Numerator: (1 +2i)(3 +4i) = 3 +4i +6i +8i² = 3 +10i -8 = -5 +10i.Denominator:9 +16=25. So z = (-5 +10i)/25= -1/5 +2/5i.Therefore, conjugate(z)= -1/5 -2/5i. So yes, this is option C.Therefore, the answer is C. I think that's solid. So I should select option C.Final Answerboxed{C}</think>Given that (i) is the imaginary unit, we need to find the conjugate of the complex number (z) that satisfies ((3 - 4i)z = 1 + 2i).First, we solve for (z) by dividing both sides of the equation by (3 - 4i):[z = frac{1 + 2i}{3 - 4i}]To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator, (3 + 4i):[z = frac{(1 + 2i)(3 + 4i)}{(3 - 4i)(3 + 4i)}]Calculating the denominator:[(3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2 = 9 + 16 = 25]Calculating the numerator:[(1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i - 8 = -5 + 10i]Thus, we have:[z = frac{-5 + 10i}{25} = frac{-1}{5} + frac{2}{5}i]The conjugate of (z) is obtained by changing the sign of the imaginary part:[overline{z} = -frac{1}{5} - frac{2}{5}i]Therefore, the conjugate of (z) is (boxed{C}).]

🔑:From (3-4i)z=1+2i, we get z= frac {1+2i}{3-4i}= frac {(1+2i)(3+4i)}{(3-4i)(3+4i)}= frac {-5+10i}{25}=- frac {1}{5}+ frac {2}{5}i,therefore bar{z}=- frac {1}{5}- frac {2}{5}i.Thus, the answer is boxed{C}.The solution simplifies the complex number using the algebraic form of complex number multiplication and division, and combines the concept of the conjugate of a complex number to find the answer. This problem tests the basic operations and concepts of complex numbers, making it a fundamental question.

🧠:[<think>Okay, let's see. I need to find the area of triangle ABC, which is a right-angled triangle at point B. They mentioned that point D is the foot of the altitude from B, so that must mean BD is perpendicular to AC. Then, they told me that AD is 3 units and DC is 4 units. Hmm. Alright, let's visualize this. So, triangle ABC with right angle at B. If I draw that out, B is the right angle, so AB and BC are the legs, and AC is the hypotenuse. Then, from B, we drop an altitude to AC, which meets AC at point D. So, BD is perpendicular to AC. Then, the segments AD and DC are given as 3 and 4, respectively. So, AD=3, DC=4, which makes the entire length of AC equal to 3 + 4 = 7. But wait, the problem isn't directly asking for AC; it's asking for the area of triangle ABC. The area of a right-angled triangle is (base * height)/2. Since it's right-angled at B, the legs AB and BC would be the base and height. If I can find the lengths of AB and BC, then multiply them and divide by 2, that should give me the area. But how do I find AB and BC? Hmm. Let me recall some properties of right-angled triangles and altitudes. In a right-angled triangle, the altitude from the right angle to the hypotenuse has some interesting properties. Specifically, the length of the altitude is the geometric mean of the lengths of the two segments it divides the hypotenuse into. Also, each leg of the triangle is the geometric mean of the hypotenuse and the adjacent segment. Wait, so maybe there's a relation here. Let me try to recall the geometric mean theorem. It states that in a right-angled triangle, the altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse. So, BD^2 = AD * DC. Given that AD is 3 and DC is 4, BD^2 should be 3*4=12, so BD= sqrt(12)=2*sqrt(3). Hmm. That might come in handy. Alternatively, the legs AB and BC can be found using the geometric mean as well. For example, AB^2 = AD * AC, and BC^2 = DC * AC. Let me check if that's correct. Wait, in the geometric mean theorem, yes, each leg is the geometric mean of the hypotenuse and the adjacent segment. So, AB is adjacent to AD, so AB^2 = AD * AC. Similarly, BC^2 = DC * AC. So, AC is 7, as we found earlier. Therefore, AB^2 = 3*7=21, so AB=sqrt(21). Similarly, BC^2=4*7=28, so BC=sqrt(28)=2*sqrt(7). Then, the area would be (AB * BC)/2. Let's compute that. sqrt(21)*sqrt(28)/2. sqrt(21*28)/2. 21*28=588. sqrt(588)/2. 588 factors into 49*12, so sqrt(49*12)=7*sqrt(12)=7*2*sqrt(3)=14*sqrt(3). Then, divide by 2, so 14*sqrt(3)/2=7*sqrt(3). Wait, but that gives me 7*sqrt(3), which is option B, but let me check again. Wait, but the area is (AB * BC)/2, right? Since AB and BC are the legs. So, if AB is sqrt(21) and BC is sqrt(28), then multiplying them gives sqrt(21*28). Let's compute 21*28. 21*28 is 588. sqrt(588). Then, simplifying sqrt(588). 588 divided by 4 is 147, so sqrt(588)=sqrt(4*147)=2*sqrt(147). Then, 147 is 49*3, so sqrt(147)=7*sqrt(3). Therefore, sqrt(588)=2*7*sqrt(3)=14*sqrt(3). So, (14*sqrt(3))/2=7*sqrt(3). So, the area would be 7*sqrt(3). But looking at the answer choices, option B is 7*sqrt(3) and option C is 21. Wait, but let me verify if this is correct. Alternatively, maybe there's another way to approach this. Since BD is the altitude, we can also compute the area as (AC * BD)/2. We know AC is 7 and BD we found as 2*sqrt(3). So, area would be (7 * 2*sqrt(3))/2 = 7*sqrt(3). Same result. So that seems to confirm the answer is 7*sqrt(3), which is option B. Wait, but hold on. Let me check the answer choices again. The options are:(A) 4√3 (B) 7√3 (C) 21 (D) 14√3 (E) 42So, 7√3 is option B, but I also got 14√3 when I first calculated sqrt(21)*sqrt(28), but then divided by 2. Wait, but according to both methods, the area is 7√3. So, maybe that's correct. But let me check again. Maybe I made a mistake in the geometric mean. Wait, let's start over. In a right-angled triangle, when an altitude is drawn from the right angle to the hypotenuse, the triangle is divided into two smaller triangles that are similar to each other and to the original triangle. Therefore, the ratios of corresponding sides should be equal. So, in triangle ABC, with right angle at B, altitude BD to hypotenuse AC. Then triangles ABD, CBD, and ABC are all similar. Therefore, ratios: In triangle ABD, sides AD=3, BD=?, AB=?In triangle CBD, sides DC=4, BD=?, BC=?In triangle ABC, sides AC=7, AB=?, BC=?Since they are similar, the ratios of corresponding sides should hold. For triangle ABD similar to triangle ABC: AD/AB = AB/AC. Therefore, AD/AB = AB/AC ⇒ AB² = AD * AC. So, AB² = 3*7=21 ⇒ AB=√21. Similarly, for triangle CBD similar to triangle ABC: DC/BC = BC/AC ⇒ BC²=DC*AC=4*7=28 ⇒ BC=√28=2√7. Therefore, the legs are √21 and 2√7. So, area is (√21 * 2√7)/2 = (2√(21*7))/2 = √147. 147=49*3, so √147=7√3. Therefore, area is 7√3. So, that's consistent. Alternatively, using the altitude BD: BD=√(AD*DC)=√(3*4)=√12=2√3. Then, area of triangle ABC is (AC * BD)/2 = (7 * 2√3)/2 = 7√3. Therefore, both methods lead to 7√3, which is answer choice B. But wait, hold on. The answer choices include 21, which is 7*3. If I miscalculated somewhere, maybe I could get 21. Let me check again. Wait, if I take BD=2√3, and then area using BD as height would be (base AC * height BD)/2 = 7*2√3 /2 = 7√3. That seems correct. Alternatively, if I use the legs AB and BC as √21 and 2√7, then area is (√21 * 2√7)/2. Let's compute that. √21 * √7 = √(21*7) = √147 = 7√3. Then multiply by 2 and divide by 2 cancels out. So, area is 7√3. Same answer. Alternatively, if I use the formula for area in terms of the segments AD and DC. Wait, is there a formula that relates AD, DC, and the area? Maybe, but I think the methods above are sufficient. Alternatively, coordinate geometry. Let me try that. Let me place point B at the origin (0,0). Since it's a right angle at B, let me set point A along the x-axis and point C along the y-axis. Then, coordinates: B(0,0), A(a,0), C(0,c). Then, the hypotenuse AC would be the line from (a,0) to (0,c). The foot of the altitude from B to AC is point D. First, find the coordinates of D. The line AC has equation (x/a) + (y/c) = 1. The altitude from B(0,0) to AC is perpendicular to AC. The slope of AC is (c - 0)/(0 - a) = -c/a. Therefore, the slope of the altitude BD is the negative reciprocal, which is a/c. So, the equation of BD is y = (a/c)x. Now, find the intersection point D between BD and AC. The equation of AC is x/a + y/c = 1. Substitute y = (a/c)x into this equation:x/a + ( (a/c)x ) /c = 1 ⇒ x/a + (a x)/c² = 1 ⇒ x (1/a + a/c²) = 1 ⇒ x = 1 / (1/a + a/c²) = 1 / ( (c² + a²)/(a c²) ) ) = (a c²)/(a² + c²).Similarly, y = (a/c)x = (a/c)*(a c²)/(a² + c²) = (a² c)/(a² + c²).Therefore, coordinates of D are ( (a c²)/(a² + c²), (a² c)/(a² + c²) ). Now, given that AD = 3 and DC = 4. The coordinates of A are (a,0), D is ( (a c²)/(a² + c²), (a² c)/(a² + c²) ). The distance AD is 3. Let's compute the distance between A and D. AD² = (a - (a c²)/(a² + c²))² + (0 - (a² c)/(a² + c²))²Simplify the x-coordinate difference: a - (a c²)/(a² + c²) = a(a² + c² - c²)/(a² + c²) ) = a(a²)/(a² + c²) ) = a³/(a² + c²)Similarly, the y-coordinate difference is - (a² c)/(a² + c²)Therefore, AD² = (a³/(a² + c²))² + ( - a² c/(a² + c²) )² = (a^6 + a^4 c²)/(a² + c²)^2 ) = a^4(a² + c²)/(a² + c²)^2 ) = a^4/(a² + c²)Similarly, DC² = 4² =16. Let's compute DC. Coordinates of D are ( (a c²)/(a² + c²), (a² c)/(a² + c²) ) and C is (0,c). DC² = (0 - (a c²)/(a² + c²))² + (c - (a² c)/(a² + c²))²First term: (a c²/(a² + c²))²Second term: c - (a² c)/(a² + c²) = c(a² + c² - a²)/(a² + c²) ) = c(c²)/(a² + c²) = c^3/(a² + c²)Therefore, DC² = (a² c^4)/(a² + c²)^2 + (c^6)/(a² + c²)^2 ) = (a² c^4 + c^6)/(a² + c²)^2 = c^4(a² + c²)/(a² + c²)^2 ) = c^4/(a² + c² )So, we have AD² = a^4/(a² + c² ) = 9 (since AD=3)And DC² = c^4/(a² + c² ) = 16 (since DC=4)Therefore, we have two equations:1) a^4/(a² + c² ) = 92) c^4/(a² + c² ) = 16Let me denote S = a² + c². Then, equations become:a^4/S =9 and c^4/S=16. Therefore, a^4 =9S and c^4=16S.Then, dividing the second equation by the first, we get (c^4)/(a^4) =16/9 ⇒ (c/a)^4 = (16/9) ⇒ (c/a)^2 =4/3 ⇒ c/a = 2/√3 or c/a = -2/√3. But lengths are positive, so c/a=2/√3 ⇒ c= (2/√3)a.Then, substitute c = (2/√3)a into S =a² +c². S= a² + (4/3)a²= (7/3)a².From first equation: a^4 =9S ⇒ a^4=9*(7/3)a² ⇒ a^4=21a² ⇒ a^4 -21a²=0 ⇒ a²(a² -21)=0 ⇒ a²=21 (since a≠0). Therefore, a=√21.Then, c= (2/√3)*a= (2/√3)*√21= 2*√(21/3)=2√7.Thus, legs AB= a=√21 and BC=c=2√7. Therefore, area=(AB*BC)/2=(√21 *2√7)/2= (√21*√7)=√147=7√3. So, same result. Therefore, the area is 7√3, which is answer choice B. But wait, looking back, the answer options include 21, which is a bigger number. Maybe I made a mistake here? Wait, 7√3 is approximately 12.12, but 21 is bigger. Let me check the answer choices again. The options are:(A) 4√3 ≈6.928(B) 7√3≈12.12(C)21(D)14√3≈24.25(E)42So, 7√3 is option B, and 21 is option C. But according to all calculations, the correct answer is 7√3. But maybe there is a miscalculation? Wait, in the coordinate geometry approach, we found AB=√21 and BC=2√7. Then, AB * BC = sqrt(21)*2*sqrt(7)=2*sqrt(147)=2*7*sqrt(3)=14*sqrt(3). Then, divide by 2 gives 7*sqrt(3). So, that's correct. Alternatively, using BD=2*sqrt(3). Then, area= (AC*BD)/2=(7*2*sqrt(3))/2=7*sqrt(3). Correct. So, both methods lead to 7*sqrt(3). So, answer is B. But why is 21 an option? Let me see. Maybe if someone thought that BD is 7, but no, BD was sqrt(12)=2*sqrt(3). Wait, or if someone thought that the area is AD*DC=3*4=12, but that's not the area. Alternatively, maybe someone confused the formula. Wait, if BD is the altitude, and the area is (AC*BD)/2=7*2*sqrt(3)/2=7*sqrt(3). If someone thought BD is 6, then 7*6/2=21, but BD is not 6. BD is sqrt(AD*DC)=sqrt(12)=2*sqrt(3). So, 7*2*sqrt(3)/2=7*sqrt(3). Alternatively, if someone thought the area is (AB*BC), not divided by 2, then sqrt(21)*2*sqrt(7)=14*sqrt(3), which is option D. But that's not the area; the area is half that. So, 14*sqrt(3)/2=7*sqrt(3). So, answer is B. Therefore, after multiple methods and checking, the answer should be B, 7√3. But wait, wait, hold on. Let me check once more. In the problem, it's a right-angled triangle at B. The altitude from B to AC is BD, which divides AC into AD=3 and DC=4. Therefore, AC=7. In a right-angled triangle, the area can be found in two ways: (AB*BC)/2 or (AC*BD)/2. So, if I can find BD, then (AC*BD)/2 would give the area. BD is the altitude. The formula for the altitude in a right-angled triangle is BD= (AB*BC)/AC. Wait, since area is (AB*BC)/2= (AC*BD)/2, so yes, BD=(AB*BC)/AC. But also, from the geometric mean, BD= sqrt(AD*DC)=sqrt(12)=2*sqrt(3). Therefore, BD=2*sqrt(3). Then, area=(AC*BD)/2=7*2*sqrt(3)/2=7*sqrt(3). So, this is another way to confirm. Alternatively, BD=2*sqrt(3). If I can find AB and BC, then multiply them and divide by 2. But since we know AD=3 and DC=4, and triangles ABD and CBD are similar to ABC, then AB= sqrt(AD*AC)=sqrt(3*7)=sqrt(21). Similarly, BC=sqrt(DC*AC)=sqrt(4*7)=sqrt(28)=2*sqrt(7). Multiply them: sqrt(21)*2*sqrt(7)=2*sqrt(147)=2*7*sqrt(3)=14*sqrt(3). Divide by 2: 7*sqrt(3). Yes, that's consistent. Therefore, all methods lead to 7*sqrt(3), which is option B. Therefore, the answer must be B. But just to be absolutely thorough, let me check for any possible mistakes. For example, did I mix up which segments correspond to the geometric mean?Wait, the geometric mean theorem (altitude-on-hypotenuse theorem) states that in a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments. So BD^2=AD*DC. So BD=sqrt(3*4)=sqrt(12)=2*sqrt(3). That's correct. Then, legs: AB^2=AD*AC=3*7=21, so AB=sqrt(21). BC^2=DC*AC=4*7=28, so BC=sqrt(28)=2*sqrt(7). Then area is (sqrt(21)*2*sqrt(7))/2= (sqrt(21*7))/1= sqrt(147)=7*sqrt(3). Yes, that's correct. So, answer is B. But let me see why option C is 21. If someone thought that the area is AD*DC=3*4=12, but that's not. Or if they thought that BD=7, but BD=2*sqrt(3). Alternatively, if someone thought that the area is (AD+DC)*something, but that's not. Alternatively, maybe miscalculating BD. But BD is indeed sqrt(12). Alternatively, if someone thought that the area is AB*BC, which is 14*sqrt(3), which is option D. But area is half that, so 7*sqrt(3). So, maybe if someone forgot to divide by 2, they would get D. But the correct answer is B. Therefore, after careful consideration, the answer is B, 7√3.Final Answerboxed{B}</think>Triangle (ABC) has a right angle at (B). Point (D) is the foot of the altitude from (B), with (AD = 3) and (DC = 4). We need to find the area of (triangle ABC).1. Hypotenuse Length: - The hypotenuse (AC = AD + DC = 3 + 4 = 7).2. Altitude Length: - The altitude (BD) from (B) to (AC) can be found using the geometric mean theorem: (BD^2 = AD cdot DC). - (BD^2 = 3 cdot 4 = 12). - (BD = sqrt{12} = 2sqrt{3}).3. Area Calculation: - The area of (triangle ABC) can be calculated using the hypotenuse and the altitude: (text{Area} = frac{1}{2} cdot AC cdot BD). - Substituting the values: (text{Area} = frac{1}{2} cdot 7 cdot 2sqrt{3} = 7sqrt{3}).4. Verification: - Using the legs (AB) and (BC) (found via geometric mean theorem): - (AB^2 = AD cdot AC = 3 cdot 7 = 21) (Rightarrow AB = sqrt{21}). - (BC^2 = DC cdot AC = 4 cdot 7 = 28) (Rightarrow BC = 2sqrt{7}). - Area calculation: (text{Area} = frac{1}{2} cdot AB cdot BC = frac{1}{2} cdot sqrt{21} cdot 2sqrt{7} = frac{1}{2} cdot 2sqrt{147} = sqrt{147} = 7sqrt{3}).Both methods confirm the area of (triangle ABC) is (7sqrt{3}).(boxed{B}),<think>Okay, so I have this problem about a right triangle ABC with a right angle at B. There's a point D, which is the foot of the altitude from B. They tell me that AD is 3 units and DC is 4 units. I need to find the area of triangle ABC. The answer choices are given, but I should try to solve it step by step first.Let me visualize the triangle. Since it's a right-angled triangle at B, I can imagine point B being the right angle, so sides AB and BC would be the legs, and AC would be the hypotenuse. Then, point D is the foot of the altitude from B to AC. So BD is perpendicular to AC, splitting AC into two segments: AD = 3 and DC = 4. Therefore, the entire length of AC is AD + DC = 3 + 4 = 7 units.Hmm, okay. So AC is 7. But how does that help me find the area? The area of triangle ABC is (AB * BC)/2. But I don't know AB or BC. However, maybe I can relate them using the altitude BD.I remember that in a right-angled triangle, the altitude to the hypotenuse has some properties. Specifically, the length of the altitude (BD) is the geometric mean of the lengths of the two segments of the hypotenuse (AD and DC). So BD = sqrt(AD * DC). Let me check if that's correct.Yes, in a right triangle, the altitude to the hypotenuse satisfies BD^2 = AD * DC. Therefore, BD = sqrt(3*4) = sqrt(12) = 2*sqrt(3). Okay, so BD is 2√3.But how does that help me find the area? Wait, the area of triangle ABC can also be calculated as (AC * BD)/2. Because AC is the hypotenuse and BD is the altitude to it. So if I can find AC and BD, then multiply them and divide by 2.But AC is 7, BD is 2√3, so area would be (7 * 2√3)/2 = 7√3. Wait, but 7√3 is one of the answer choices, option B. But hold on, the answer choices also include 21 and 42. Hmm, maybe I made a mistake here.Alternatively, maybe I should compute the area using the legs AB and BC. Let me think. Let me recall another property: in a right-angled triangle, the legs are the geometric mean of the hypotenuse and the adjacent segment. So, AB^2 = AD * AC and BC^2 = DC * AC. Let me verify.Yes, that's another property. So, AB^2 = AD * AC = 3 * 7 = 21, so AB = sqrt(21). Similarly, BC^2 = DC * AC = 4 * 7 = 28, so BC = sqrt(28) = 2*sqrt(7). Then the area would be (AB * BC)/2 = (sqrt(21) * 2*sqrt(7))/2.Multiplying sqrt(21) and sqrt(7) gives sqrt(147). Because sqrt(21)*sqrt(7) = sqrt(21*7) = sqrt(147). Then, 2*sqrt(147)/2 = sqrt(147). Simplify sqrt(147): 147 = 49*3, so sqrt(49*3)=7*sqrt(3). So the area is 7*sqrt(3), which matches the previous result. So both methods give me 7√3, which is option B.But hold on, the answer choice C is 21, which is different. Let me double-check my steps because maybe I made a mistake here.Wait, so BD is the altitude to hypotenuse AC. So BD = sqrt(AD * DC) = sqrt(12) = 2*sqrt(3). Then area is (AC * BD)/2 = (7 * 2*sqrt(3))/2 = 7*sqrt(3). That seems correct.Alternatively, using the legs: AB = sqrt(AD * AC) = sqrt(3*7) = sqrt(21). BC = sqrt(DC * AC) = sqrt(4*7) = sqrt(28) = 2*sqrt(7). Then area is (sqrt(21) * 2*sqrt(7))/2 = (sqrt(21*7) * 2)/2 = sqrt(147) = 7*sqrt(3). So that's consistent.But then why is 21 an option? Let me check if the problem is stated correctly. It says AD = 3 and DC = 4. Then AC = 7. So that's correct. The area via BD is 7*sqrt(3), which is answer B. However, let me check once more because 21 is tempting.Wait, maybe I misapplied the formula. Let me recall the formula for the area. If BD is the altitude to AC, then area is ½ * AC * BD. So AC is 7, BD is 2*sqrt(3), so ½ *7 *2*sqrt(3) is 7*sqrt(3). That's right. So 7*sqrt(3) is the area.Alternatively, perhaps the problem is not using the altitude but something else? Wait, but the problem states D is the foot of the altitude from B, so BD is perpendicular to AC. So the standard formulas should apply.Alternatively, maybe there's a different approach. Let me try to use coordinate geometry. Let's place point B at (0,0), since it's the right angle. Let me assume point A is on the x-axis and point C is on the y-axis. Wait, but if D is the foot of the altitude from B to AC, then BD is perpendicular to AC. But if we place B at (0,0), A at (a,0), and C at (0,c), then AC is the hypotenuse from (a,0) to (0,c). The foot of the perpendicular from B to AC would be a point D.But this might complicate things. Alternatively, maybe it's better to use coordinate geometry with AC as the x-axis. Wait, but then B would be somewhere above AC. Let me try.Let me place AC horizontally, with point A at (0,0), point C at (7,0), so AC is from (0,0) to (7,0). Then point D is somewhere on AC. AD is 3, so D is at (3,0). DC is 4, so from (3,0) to (7,0). Wait, but BD is the altitude from B to AC, so BD is vertical? If AC is horizontal, then BD would be vertical. But then point B would be at (3, h) for some h. Then, triangle ABC has coordinates A(0,0), B(3,h), C(7,0). Then AB is from (0,0) to (3,h), BC is from (3,h) to (7,0), and since ABC is a right-angled triangle at B, the legs are AB and BC, and the hypotenuse is AC.Wait, but in this coordinate system, AC is the hypotenuse, and ABC is right-angled at B. So AB and BC should be perpendicular. Let's check the vectors. Vector AB is (3, h) and vector BC is (4, -h). The dot product should be zero. So (3)(4) + (h)(-h) = 12 - h² = 0. Therefore, h² = 12, so h = 2√3. Therefore, point B is at (3, 2√3). Then, the area of triangle ABC is (AB * BC)/2. Let's compute AB and BC.AB is the distance from A(0,0) to B(3,2√3): sqrt(3² + (2√3)^2) = sqrt(9 + 12) = sqrt(21). BC is the distance from B(3,2√3) to C(7,0): sqrt((4)^2 + (-2√3)^2) = sqrt(16 + 12) = sqrt(28) = 2√7. Then the area is (sqrt(21) * 2√7)/2 = (2√147)/2 = sqrt(147) = 7√3. So again, same result. So this method also gives 7√3.Alternatively, since in coordinate geometry, the area can also be computed as the determinant. Coordinates: A(0,0), B(3,2√3), C(7,0). The area is ½ | (3)(0 - 2√3) + 7(2√3 - 0) + 0(0 - 0) |. Wait, that's ½ | 3*(-2√3) + 7*(2√3) | = ½ | -6√3 + 14√3 | = ½ |8√3| = 4√3. Wait, that contradicts the previous results. What did I do wrong here?Wait, no. The formula for the area using coordinates is ½ | x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B) |. Plugging in A(0,0), B(3,2√3), C(7,0):Area = ½ | 0*(2√3 - 0) + 3*(0 - 0) + 7*(0 - 2√3) | = ½ | 0 + 0 + 7*(-2√3) | = ½ | -14√3 | = ½ *14√3 = 7√3. Ah, that's correct. I must have miscalculated earlier. So it is indeed 7√3. So the coordinate geometry method also confirms that. Therefore, answer B is correct.But why is 21 an option? Maybe if someone thought BD is 7 and multiplied by AC/2? Wait, 7*6/2=21. But BD is not 6. Alternatively, maybe some other miscalculation. But according to all methods I tried, 7√3 is the area.Wait, perhaps I made a mistake in the problem statement. Let me check again. The problem says triangle ABC has a right angle at B. Point D is the foot of the altitude from B. AD=3, DC=4. So AC=7. Then BD=sqrt(AD*DC)=sqrt(12)=2sqrt(3). Then area is (AC*BD)/2= (7*2sqrt(3))/2=7sqrt(3). So that's correct.Alternatively, maybe someone thought that AD=3 and DC=4, so BD is the geometric mean, which is sqrt(3*4)=2sqrt(3). Then, area of ABC is (AB * BC)/2. But AB and BC can be found via similar triangles.Wait, triangles ABD and CBD are similar to ABC and each other. So ratios apply. So in triangle ABD, AD/AB = AB/AC, so AB^2 = AD*AC=3*7=21, so AB=sqrt(21). Similarly, in triangle CBD, DC/BC = BC/AC, so BC^2=DC*AC=4*7=28, so BC=sqrt(28). Then area is (sqrt(21)*sqrt(28))/2 = sqrt(588)/2 = sqrt(49*12)/2 = (7*2*sqrt(3))/2=7sqrt(3). So same answer.Therefore, all approaches lead to 7√3, which is option B. So I think that is correct. The answer choices must have some distractors, but B is the right one.Final Answerboxed{B}</think>Triangle (ABC) has a right angle at (B). Point (D) is the foot of the altitude from (B) to (AC), with (AD = 3) and (DC = 4). We need to find the area of (triangle ABC).1. Length of (AC): [ AC = AD + DC = 3 + 4 = 7 ]2. Length of the altitude (BD): Using the geometric mean property for the altitude in a right triangle: [ BD^2 = AD times DC implies BD^2 = 3 times 4 = 12 implies BD = 2sqrt{3} ]3. Area of (triangle ABC): The area can be calculated using the hypotenuse (AC) and the altitude (BD): [ text{Area} = frac{1}{2} times AC times BD = frac{1}{2} times 7 times 2sqrt{3} = 7sqrt{3} ]4. Verification using coordinate geometry: Placing points (A(0,0)), (C(7,0)), and (D(3,0)), we find (B(3, 2sqrt{3})) by ensuring the dot product of vectors (AB) and (BC) is zero. The area calculation confirms the result: [ text{Area} = frac{1}{2} left| 0 + 0 + 7(-2sqrt{3}) right| = frac{1}{2} times 14sqrt{3} = 7sqrt{3} ]Thus, the area of (triangle ABC) is (boxed{B}).]

🔑:It is a well-known fact that in any right triangle ABC with the right angle at B and D the foot of the altitude from B onto AC we have BD^2 = ADcdot CD. (See below for a proof.) Then BD = sqrt{ 3cdot 4 } = 2sqrt 3, and the area of the triangle ABC is frac{ACcdot BD}2 = 7sqrt3Rightarrowboxed{text{(B)}}.Proof: Consider the [Pythagorean theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_theorem) for each of the triangles ABC, ABD, and CBD. We get:AB^2 + BC^2 = AC^2 = (AD+DC)^2 = AD^2 + DC^2 + 2 cdot AD cdot DC.AB^2 = AD^2 + BD^2BC^2 = BD^2 + CD^2Substituting equations 2 and 3 into the left hand side of equation 1, we get BD^2 = AD cdot DC. Alternatively, note that triangle ABD sim triangle BCD Longrightarrow frac{AD}{BD} = frac{BD}{CD}.

🧠:[<think>Okay, so I need to find the limit of this sequence as n approaches infinity: (cube root of n squared minus square root of n squared plus 5) divided by (fifth root of n to the seventh power minus square root of n plus 1). Hmm, let me write that down properly to make sure I understand.The sequence is:lim(n→∞) [n^(2/3) - sqrt(n² + 5)] / [n^(7/5) - sqrt(n + 1)]Alright, so both the numerator and the denominator have terms with roots, which means exponents that are fractions. Since it's a limit as n approaches infinity, I should probably look at the dominant terms in the numerator and the denominator. Dominant terms are the ones that grow the fastest as n gets very large.First, let's analyze the numerator: n^(2/3) minus sqrt(n² + 5). Let me rewrite sqrt(n² + 5) as (n² + 5)^(1/2). When n is very large, the 5 inside the square root becomes negligible, so sqrt(n² + 5) ≈ sqrt(n²) = n. So the numerator is approximately n^(2/3) - n. But since n grows much faster than n^(2/3), the dominant term here should be -n.Wait, but let me check that again. The numerator is n^(2/3) - sqrt(n² +5). sqrt(n² +5) is indeed approximately n for large n, so the numerator behaves like n^(2/3) - n. So as n becomes very large, the term with the higher exponent will dominate. Since n is the same as n^(1), and 1 is larger than 2/3, the dominant term is -n. So the numerator behaves like -n as n approaches infinity.Now the denominator: n^(7/5) - sqrt(n + 1). Let's rewrite sqrt(n +1) as (n +1)^(1/2). For large n, the 1 is negligible, so sqrt(n +1) ≈ sqrt(n) = n^(1/2). Therefore, the denominator is approximately n^(7/5) - n^(1/2). Let's compare the exponents: 7/5 is equal to 1.4, and 1/2 is 0.5. So n^(7/5) grows much faster than n^(1/2). Therefore, the dominant term in the denominator is n^(7/5).So putting it all together, the expression behaves like (-n) / (n^(7/5)) as n approaches infinity. Let's simplify that: -n / n^(7/5) = -n^(1 - 7/5) = -n^(-2/5) = -1 / n^(2/5). As n approaches infinity, n^(2/5) also approaches infinity, so the whole expression tends to 0. Therefore, the limit should be 0.But wait, let me verify this step by step more carefully because sometimes even if the dominant terms suggest a certain behavior, other terms might contribute in a way that affects the limit. Maybe I need to use a more precise expansion.Starting with the numerator: n^(2/3) - sqrt(n² +5). Let's factor out the dominant term from sqrt(n² +5). So sqrt(n² +5) = n * sqrt(1 + 5/n²). Using the Taylor expansion for sqrt(1 + x) ≈ 1 + x/2 - x²/8 + ... for small x. Here, x = 5/n², which tends to 0 as n approaches infinity. So sqrt(1 + 5/n²) ≈ 1 + (5)/(2n²). Therefore, sqrt(n² +5) ≈ n*(1 + 5/(2n²)) = n + 5/(2n).So the numerator becomes n^(2/3) - [n + 5/(2n)] = -n + n^(2/3) - 5/(2n). As n becomes large, the leading term is -n, followed by n^(2/3), and then the term -5/(2n) which is negligible. But even so, the dominant term is -n. So the numerator is approximately -n for large n.Similarly, let's handle the denominator: n^(7/5) - sqrt(n +1). For sqrt(n +1), let's factor out n: sqrt(n +1) = sqrt(n*(1 + 1/n)) = sqrt(n)*sqrt(1 + 1/n) ≈ sqrt(n)*(1 + 1/(2n) - 1/(8n²) + ...). So sqrt(n +1) ≈ sqrt(n) + 1/(2sqrt(n)) - 1/(8n^(3/2)) + ... Therefore, the denominator is n^(7/5) - [sqrt(n) + 1/(2sqrt(n)) - ...] ≈ n^(7/5) - n^(1/2). As before, since 7/5 = 1.4 is greater than 1/2 = 0.5, the dominant term is n^(7/5). So the denominator is approximately n^(7/5).Therefore, the expression is approximately (-n)/n^(7/5) = -n^(1 - 7/5) = -n^(-2/5) = -1/n^(2/5). As n approaches infinity, n^(2/5) approaches infinity, so -1/n^(2/5) approaches 0. Therefore, the limit should be 0.But let me check this again. Maybe there's a mistake in the approximation steps. Let's go through numerator and denominator again with more precision.Numerator: n^(2/3) - sqrt(n² + 5)Express sqrt(n² +5) as n*sqrt(1 + 5/n²). Then use the expansion sqrt(1 + x) ≈ 1 + (x)/2 - (x²)/8 + ... for x near 0. Here, x = 5/n², so:sqrt(n² +5) = n*(1 + (5)/(2n²) - (25)/(8n^4) + ...) = n + 5/(2n) - 25/(8n^3) + ...Therefore, numerator: n^(2/3) - [n + 5/(2n) - ...] = -n + n^(2/3) -5/(2n) + ... So the leading terms are -n + n^(2/3). So the dominant term is -n, and the next term is n^(2/3). Since n^(2/3) is much smaller than n for large n, the numerator is dominated by -n.Denominator: n^(7/5) - sqrt(n +1)sqrt(n +1) can be written as sqrt(n)*sqrt(1 + 1/n) ≈ sqrt(n)*(1 + 1/(2n) - 1/(8n²) + ...) = sqrt(n) + 1/(2n^(1/2)) - 1/(8n^(3/2)) + ... So the denominator is n^(7/5) - sqrt(n) - 1/(2n^(1/2)) + ... The leading terms are n^(7/5) - n^(1/2). Since 7/5 is 1.4 and 1/2 is 0.5, n^(7/5) is dominant.Therefore, denominator is approximately n^(7/5).Thus, the entire expression is approximately (-n)/n^(7/5) = -n^(1 - 7/5) = -n^(-2/5) = -1/n^(2/5), which tends to 0 as n approaches infinity. Therefore, the limit is 0.Wait, but let me confirm with another approach. Maybe divide numerator and denominator by the highest power of n in the denominator.Alternatively, factor out the dominant terms from numerator and denominator.In the numerator: the dominant term is -n. Let's factor out n from the numerator:Numerator: n^(2/3) - sqrt(n² +5) = -n + n^(2/3) - 5/(2n) ≈ -n[1 - n^(2/3 -1) + ...] Hmm, but maybe factoring n is messy here.Alternatively, in the numerator, the dominant term is -n, and in the denominator, it's n^(7/5). So the ratio is (-n)/n^(7/5) = -n^(-2/5) as before. So as n approaches infinity, this goes to 0. So the limit is 0.Alternatively, another way to think about it is to divide numerator and denominator by n^(7/5), which is the dominant term in the denominator.Let me try that.Divide numerator and denominator by n^(7/5):Numerator: [n^(2/3) - sqrt(n² +5)] / n^(7/5) = n^(2/3)/n^(7/5) - sqrt(n² +5)/n^(7/5) = n^(2/3 -7/5) - sqrt(n² +5)/n^(7/5)Calculate exponents:2/3 -7/5 = (10 -21)/15 = -11/15. So first term is n^(-11/15) = 1/n^(11/15)Second term: sqrt(n² +5)/n^(7/5) = n*sqrt(1 +5/n²)/n^(7/5) = n^(1 -7/5) * sqrt(1 +5/n²) = n^(-2/5)*sqrt(1 +5/n²) ≈ n^(-2/5)*(1 + 5/(2n²)) ≈ n^(-2/5) as n becomes large.Therefore, numerator divided by n^(7/5) is approximately 1/n^(11/15) - n^(-2/5). Since 11/15 ≈ 0.733 and 2/5 = 0.4. So 1/n^(11/15) tends to 0 and -n^(-2/5) tends to 0, but which one is bigger?Wait, but 11/15 is larger than 2/5, so 1/n^(11/15) tends to 0 faster than n^(-2/5). Therefore, the dominant term here is -n^(-2/5).Similarly, denominator divided by n^(7/5) is [n^(7/5) - sqrt(n +1)]/n^(7/5) = 1 - sqrt(n +1)/n^(7/5) = 1 - n^(1/2)/n^(7/5) = 1 - n^(1/2 -7/5) = 1 - n^(-9/10). As n approaches infinity, n^(-9/10) tends to 0, so denominator divided by n^(7/5) tends to 1.Therefore, the entire expression becomes [ -n^(-2/5) + ... ] / [1 - n^(-9/10) + ...] ≈ (-n^(-2/5))/1 = -n^(-2/5) → 0 as n→infty.Therefore, the limit is 0.Alternatively, another approach is to use the leading terms. The highest power in the numerator is n (from the sqrt(n² +5)), and the highest power in the denominator is n^(7/5). So the ratio is n / n^(7/5) = n^(1 -7/5) = n^(-2/5) → 0. But in the numerator, it's -n, so the ratio is -n^(-2/5) → 0.Hence, regardless of the method, the limit is 0.Wait, but in the numerator, the leading term is -n, which is negative, and the denominator is positive (since n^(7/5) is positive and much larger than sqrt(n +1)), so the entire expression is negative, approaching 0 from the negative side. But the limit is still 0.Therefore, the final answer should be 0.I think this is correct, but let me check with another example. Suppose I have similar terms, like lim(n→∞) [n^(1/2) - n]/[n^2 - n^(1/3)]. The numerator's dominant term is -n, denominator's dominant term is n^2, so ratio is -n /n^2 = -1/n → 0. So similar logic applies here.Therefore, in our problem, same idea. Dominant terms in numerator and denominator are -n and n^(7/5), so ratio is -n/n^(7/5) = -1/n^(2/5) → 0. So yes, the limit is 0.Hence, the answer is 0.Final Answerboxed{0}</think>To find the limit of the sequence:[lim _{n rightarrow infty} frac{sqrt[3]{n^{2}}-sqrt{n^{2}+5}}{sqrt[5]{n^{7}}-sqrt{n+1}}]we analyze the dominant terms in the numerator and the denominator as ( n ) approaches infinity.Numerator Analysis:- The term ( sqrt[3]{n^{2}} ) is ( n^{2/3} ).- The term ( sqrt{n^{2}+5} ) approximates ( n ) for large ( n ) because the ( +5 ) becomes negligible.- Therefore, the numerator ( n^{2/3} - sqrt{n^{2}+5} ) behaves like ( n^{2/3} - n ). The dominant term here is ( -n ) since ( n ) grows much faster than ( n^{2/3} ).Denominator Analysis:- The term ( sqrt[5]{n^{7}} ) is ( n^{7/5} ).- The term ( sqrt{n+1} ) approximates ( n^{1/2} ) for large ( n ).- Therefore, the denominator ( sqrt[5]{n^{7}} - sqrt{n+1} ) behaves like ( n^{7/5} - n^{1/2} ). The dominant term here is ( n^{7/5} ) since ( n^{7/5} ) grows much faster than ( n^{1/2} ).Simplifying the Expression:- The dominant terms give us the approximation ( frac{-n}{n^{7/5}} ).- Simplifying this ratio: ( frac{-n}{n^{7/5}} = -n^{1 - 7/5} = -n^{-2/5} = -frac{1}{n^{2/5}} ).As ( n ) approaches infinity, ( n^{2/5} ) also approaches infinity, making ( -frac{1}{n^{2/5}} ) approach 0.Thus, the limit of the sequence is:[boxed{0}],<think>Okay, so I need to find the limit of this sequence as n approaches infinity. The expression is (cube root of n squared minus square root of n squared plus 5) divided by (fifth root of n^7 minus square root of n plus 1). Hmm, that looks a bit complicated, but maybe I can simplify each term by expressing them with exponents instead of roots. Let me write that down.First, let's rewrite all the terms using exponents. The cube root of n squared is n^(2/3). The square root of n squared plus 5 is sqrt(n² + 5), which can be written as (n² + 5)^(1/2). The denominator has the fifth root of n^7, which is n^(7/5), and the square root of n plus 1 is (n + 1)^(1/2). So the whole expression becomes:[n^(2/3) - (n² + 5)^(1/2)] / [n^(7/5) - (n + 1)^(1/2)]Now, I need to analyze the behavior of each term as n approaches infinity. For large n, the dominant terms will be the ones with the highest powers, right? So maybe I can approximate each part by ignoring the lower order terms.Starting with the numerator: n^(2/3) minus (n² + 5)^(1/2). Let's look at each term. The first term, n^(2/3), grows as n increases, but the second term, (n² + 5)^(1/2), simplifies to approximately n when n is very large because the 5 becomes negligible. So sqrt(n² + 5) ≈ sqrt(n²) = n. Therefore, the numerator is roughly n^(2/3) - n.Similarly, in the denominator: n^(7/5) minus sqrt(n + 1). The term n^(7/5) is n^(1.4), and sqrt(n + 1) is approximately sqrt(n) for large n. So the denominator is approximately n^(7/5) - n^(1/2).So now the expression is approximately (n^(2/3) - n) / (n^(7/5) - n^(1/2)). To find the limit, we can factor out the dominant terms from numerator and denominator.In the numerator, the dominant term is -n (since n grows much faster than n^(2/3)). So we can factor out n from the numerator:n^(2/3) - n = n^(2/3) - n = -n + n^(2/3) = -n[1 - n^(2/3 - 1)] = -n[1 - n^(-1/3)]. Wait, maybe that's a bit messy. Alternatively, factor out the highest power term. Let me think.Wait, in the numerator, as n approaches infinity, the term -n dominates over n^(2/3). So when factoring, the leading term is -n, so we can write the numerator as -n[1 - n^(2/3)/n] = -n[1 - n^(-1/3)]. Similarly, the denominator has n^(7/5) as the dominant term, so factor that out:n^(7/5) - n^(1/2) = n^(7/5)[1 - n^(1/2 - 7/5)] = n^(7/5)[1 - n^(-9/10)].So substituting back into the expression, we have:[-n[1 - n^(-1/3)]] / [n^(7/5)[1 - n^(-9/10)]] = (-n / n^(7/5)) * [1 - n^(-1/3)] / [1 - n^(-9/10)]Simplify the exponent in the first fraction: n / n^(7/5) = n^(1 - 7/5) = n^(-2/5) = 1 / n^(2/5). So now we have:(-1 / n^(2/5)) * [1 - n^(-1/3)] / [1 - n^(-9/10)]Now, as n approaches infinity, the terms n^(-1/3) and n^(-9/10) both approach zero. Therefore, [1 - n^(-1/3)] approaches 1, and [1 - n^(-9/10)] also approaches 1. So the entire expression simplifies to (-1 / n^(2/5)) * 1 / 1 = -1 / n^(2/5). As n approaches infinity, n^(2/5) approaches infinity, so -1 / n^(2/5) approaches 0. Therefore, the limit is 0. Wait, but let me check this again because sometimes when dealing with limits, even if individual terms approach zero, the overall expression might not. Hmm.Wait a second, maybe I made a mistake in factoring. Let me go back. Let's re-examine the numerator and denominator.Numerator: n^(2/3) - sqrt(n² + 5). As n becomes very large, sqrt(n² + 5) is approximately n, but more precisely, sqrt(n² + 5) = n * sqrt(1 + 5/n²) ≈ n(1 + (5)/(2n²)) using the binomial approximation for sqrt(1 + x) ≈ 1 + x/2 when x is small. So sqrt(n² +5) ≈ n + 5/(2n). Therefore, the numerator becomes n^(2/3) - [n + 5/(2n)].So numerator ≈ n^(2/3) - n - 5/(2n). Similarly, the dominant term here is -n. So when we factor out -n, we get:-n [1 - n^(2/3)/n + 5/(2n^2)/n] = -n [1 - n^(-1/3) + 5/(2n^3)].But maybe instead of factoring out, we can compare the leading terms. So numerator is dominated by -n, denominator is dominated by n^(7/5). Therefore, the ratio is (-n)/n^(7/5) = -n^(1 - 7/5) = -n^(-2/5) = -1/n^(2/5). Then as n approaches infinity, this term goes to 0. So the limit is 0.But let me check the denominator as well. The denominator is n^(7/5) - sqrt(n + 1). sqrt(n + 1) is approximately sqrt(n) for large n, so sqrt(n) = n^(1/2). So the denominator is n^(7/5) - n^(1/2). Now, n^(7/5) is equal to n^(1.4), and n^(1/2) is n^(0.5). So n^(1.4) is much larger than n^(0.5) as n approaches infinity, so the denominator is dominated by n^(7/5). Therefore, denominator ≈ n^(7/5). Therefore, the overall expression is approximately (-n)/n^(7/5) = -n^(-2/5), which tends to 0 as n approaches infinity. Therefore, the limit should be 0.But let me verify this by another method. Maybe divide numerator and denominator by the highest power of n in the denominator. Wait, the denominator's highest power is n^(7/5). So if we divide numerator and denominator by n^(7/5), let's see:Numerator: [n^(2/3) - sqrt(n² + 5)] / n^(7/5) = n^(2/3)/n^(7/5) - sqrt(n² + 5)/n^(7/5)Denominator: [n^(7/5) - sqrt(n + 1)] / n^(7/5) = 1 - sqrt(n + 1)/n^(7/5)So the entire expression becomes:[ n^(-11/15) - sqrt(n² +5)/n^(7/5) ] / [1 - sqrt(n +1)/n^(7/5) ]Let's compute each term:First term in numerator: n^(2/3 - 7/5) = n^(10/15 - 21/15) = n^(-11/15). As n approaches infinity, n^(-11/15) approaches 0.Second term in numerator: sqrt(n² +5)/n^(7/5) = [n * sqrt(1 + 5/n²)] / n^(7/5) = n / n^(7/5) * sqrt(1 + 0) = n^(1 - 7/5) = n^(-2/5) = 1/n^(2/5). So sqrt(n² +5)/n^(7/5) ≈ 1/n^(2/5) as n approaches infinity, which tends to 0.Denominator terms: sqrt(n +1)/n^(7/5) ≈ sqrt(n)/n^(7/5) = n^(1/2)/n^(7/5) = n^(1/2 - 7/5) = n^(-9/10), which also tends to 0 as n approaches infinity. Therefore, denominator is 1 - 0 = 1.So putting it all together, the numerator becomes 0 - 0 = 0, and the denominator is 1, so the limit is 0/1 = 0. Therefore, the limit is indeed 0.Wait, but let me check once more to make sure. Alternatively, perhaps using L’Hospital’s Rule? But since this is a sequence, we can consider the function as x approaches infinity and apply L’Hospital’s Rule, then convert back to the sequence. Let me try that.Let me set x instead of n, so we have the function f(x) = [x^(2/3) - sqrt(x² +5)] / [x^(7/5) - sqrt(x +1)]. Then, as x approaches infinity, we can see if L’Hospital’s Rule applies. But for L’Hospital, both numerator and denominator need to approach infinity or 0. Let's check:Numerator as x approaches infinity: x^(2/3) - sqrt(x² +5) ≈ x^(2/3) - x, which tends to -infinity.Denominator: x^(7/5) - sqrt(x +1) ≈ x^(7/5) - x^(1/2), which tends to infinity because x^(7/5) is higher order. So both numerator and denominator tend to -infinity and infinity, respectively. So the form is (-infinity)/infinity, which is equivalent to -infinity/infinity. So we can apply L’Hospital’s Rule. Taking derivative of numerator and denominator.Derivative of numerator: d/dx [x^(2/3) - sqrt(x² +5)] = (2/3)x^(-1/3) - (1/(2sqrt(x² +5)))(2x) = (2)/(3x^(1/3)) - x / sqrt(x² +5)Derivative of denominator: d/dx [x^(7/5) - sqrt(x +1)] = (7/5)x^(2/5) - (1)/(2sqrt(x +1))Now, let's evaluate these derivatives as x approaches infinity.First, the derivative of the numerator:(2)/(3x^(1/3)) approaches 0.x / sqrt(x² +5) = x / (x sqrt(1 + 5/x²)) = 1 / sqrt(1 + 5/x²) ≈ 1 - (5)/(2x²) as x approaches infinity. Therefore, x / sqrt(x² +5) approaches 1. Therefore, the derivative of numerator approaches 0 - 1 = -1.Derivative of denominator:(7/5)x^(2/5) approaches infinity as x approaches infinity.(1)/(2sqrt(x +1)) approaches 0. So the derivative of denominator approaches (7/5)x^(2/5) - 0, which goes to infinity.Therefore, applying L’Hospital’s Rule, the limit becomes (-1)/infinity = 0.Therefore, the limit is 0. That's the same result as before.Alternatively, another approach: Let's divide numerator and denominator by n, since both numerator and denominator have terms that can be related to n.Wait, but let's see. The numerator is approximately -n, the denominator is approximately n^(7/5). So dividing numerator and denominator by n gives (-1) / [n^(7/5)/n] = (-1)/n^(2/5) which tends to 0. Same result.So all approaches seem to confirm that the limit is 0. Therefore, the answer should be 0. Let me check once again for any possible miscalculations.Wait, perhaps I should check with specific large values of n. Let's take n = 10^15, a very large number.Compute numerator: cube root of (10^15)^2 = (10^30)^(1/3) = 10^10. sqrt((10^15)^2 +5) = sqrt(10^30 +5) ≈ 10^15. Therefore, numerator ≈ 10^10 - 10^15 ≈ -10^15.Denominator: fifth root of (10^15)^7 = (10^105)^(1/5) = 10^21. sqrt(10^15 +1) ≈ 10^7.5. So denominator ≈ 10^21 - 10^7.5 ≈ 10^21.Therefore, the ratio is approximately (-10^15)/10^21 = -10^(-6) = -0.000001, which is approaching 0. As n increases further, this value gets even smaller, approaching 0. So numerical example supports the analytical result.Therefore, I can be confident that the limit is 0.Final Answerboxed{0}</think>To find the limit of the sequence (lim _{n rightarrow infty} frac{sqrt[3]{n^{2}}-sqrt{n^{2}+5}}{sqrt[5]{n^{7}}-sqrt{n+1}}), we start by rewriting each term using exponents:[frac{n^{2/3} - (n^2 + 5)^{1/2}}{n^{7/5} - (n + 1)^{1/2}}]For large (n), we approximate the terms by their dominant behavior. The numerator (n^{2/3} - sqrt{n^2 + 5}) is dominated by (-n) because (sqrt{n^2 + 5} approx n). The denominator (n^{7/5} - sqrt{n + 1}) is dominated by (n^{7/5}) because (sqrt{n + 1} approx sqrt{n}).Thus, the expression simplifies to:[frac{-n}{n^{7/5}}]Simplifying the exponent:[frac{-n}{n^{7/5}} = -n^{1 - 7/5} = -n^{-2/5} = -frac{1}{n^{2/5}}]As (n rightarrow infty), (n^{2/5} rightarrow infty), so (-frac{1}{n^{2/5}} rightarrow 0).To confirm, we use L'Hospital's Rule on the function (f(x) = frac{x^{2/3} - sqrt{x^2 + 5}}{x^{7/5} - sqrt{x + 1}}). The derivatives of the numerator and denominator as (x rightarrow infty) are (-1) and (infty) respectively, leading to the limit (0).Thus, the limit is:[boxed{0}]]

🔑:## Solutionbegin{aligned}& lim _{n rightarrow infty} frac{sqrt[3]{n^{2}}-sqrt{n^{2}+5}}{sqrt[5]{n^{7}}-sqrt{n+1}}=lim _{n rightarrow infty} frac{n^{-frac{7}{5}}left(sqrt[3]{n^{2}}-sqrt{n^{2}+5}right)}{n^{-frac{7}{5}}left(sqrt[5]{n^{7}}-sqrt{n+1}right)}= & =lim _{n rightarrow infty} frac{sqrt[3]{n^{2} cdot n^{-frac{21}{5}}}-sqrt{n^{2} cdot n^{-frac{14}{5}}+5 n^{-frac{14}{5}}}}{sqrt[5]{n^{7} cdot n^{-7}}-sqrt{n cdot n^{-frac{14}{5}}}+n^{-frac{14}{5}}}= & =lim _{n rightarrow infty} frac{sqrt[3]{n^{-frac{11}{5}}}-sqrt{n^{-frac{4}{5}}+5 n^{-frac{14}{5}}}}{sqrt[5]{1}-sqrt{n^{-frac{9}{5}}+n^{-frac{14}{5}}}}= & =frac{sqrt[3]{0}-sqrt{0+5 cdot 0}}{1-sqrt{0+0}}=frac{0}{1}=0end{aligned}Problem Kuznetsov Limits 4-22

🧠:[<think>Okay, let me try to solve this problem. Let's read it again carefully to make sure I understand all the details.We need to divide 40 chess pieces into 27 piles. Each pile can have 1, 2, or 3 chess pieces. Additionally, the number of piles with only 1 chess piece is twice the number of the remaining piles. We need to find the number of piles with exactly 2 chess pieces.Hmm, let me break this down. So, first, there are 27 piles total. Each pile has either 1, 2, or 3 pieces. The total number of pieces is 40. The key condition here is that the number of piles with 1 piece is twice the number of the remaining piles. The "remaining piles" would be those that don't have 1 piece, right? So the piles with 2 or 3 pieces. Let me confirm that.Let me parse the sentence again: "the number of piles with only 1 chess piece is twice the number of the remaining piles". So "remaining piles" are the ones that are not in the group with 1 piece. So if there are x piles with 1 piece, then the remaining piles are 27 - x. But according to the problem, x is twice the number of remaining piles. Wait, that seems conflicting. If x is twice the remaining piles, but remaining piles are 27 - x. So x = 2*(27 - x). Let me solve this equation.x = 2*(27 - x)x = 54 - 2x3x = 54x = 18.So, the number of piles with 1 piece is 18. Therefore, the remaining piles are 27 - 18 = 9. So there are 9 piles that have either 2 or 3 pieces.So now, we have 18 piles with 1 piece each, 9 piles with either 2 or 3 pieces. Let's denote the number of piles with 2 pieces as y, and the number with 3 pieces as z. Then we know that:y + z = 9 (since there are 9 remaining piles)Also, the total number of chess pieces is 40. The total from the 1-piece piles is 18*1 = 18. The total from the 2 and 3-piece piles is 2y + 3z. So the equation is:18 + 2y + 3z = 40Simplify that:2y + 3z = 40 - 18 = 22So now we have two equations:1. y + z = 92. 2y + 3z = 22We can solve this system of equations. Let's solve for one variable in terms of the other. From equation 1, y = 9 - z. Substitute into equation 2:2(9 - z) + 3z = 2218 - 2z + 3z = 2218 + z = 22z = 22 - 18 = 4Then y = 9 - z = 9 - 4 = 5So, the number of piles with exactly 2 chess pieces is 5.Wait, let me check my calculations again to be sure.First, we have 18 piles with 1 piece, which uses 18 pieces. The remaining 40 - 18 = 22 pieces must be in the 9 piles of 2 or 3. So, if there are y piles of 2 and z piles of 3, then:y + z = 92y + 3z = 22Solving:From the first equation, z = 9 - y. Substitute into the second equation:2y + 3(9 - y) = 222y + 27 - 3y = 22- y + 27 = 22- y = -5y = 5Then z = 9 - 5 = 4So that's 5 piles of 2 and 4 piles of 3. Let's check the total pieces:18*1 + 5*2 + 4*3 = 18 + 10 + 12 = 40. Yes, that adds up. And the number of piles is 18 + 5 + 4 = 27. Correct.Therefore, the answer should be 5.Wait, but let me just make sure there are no other interpretations of the problem. The problem says "the number of piles with only 1 chess piece is twice the number of the remaining piles". So remaining piles are those that are not with 1, i.e., 2 or 3. So the problem's wording seems correct here. So the 18 piles with 1 are twice the remaining 9. 18 is twice 9. That's correct.So, I think the answer is 5. So the number of piles with exactly 2 chess pieces is 5.Final Answerboxed{5}</think>We need to divide 40 chess pieces into 27 piles where each pile contains 1, 2, or 3 pieces. The number of piles with 1 piece is twice the number of remaining piles (which have 2 or 3 pieces).First, we determine the number of piles with 1 piece. Let ( x ) be the number of piles with 1 piece. The remaining piles are ( 27 - x ). According to the problem, ( x = 2(27 - x) ):[begin{align*}x &= 2(27 - x) x &= 54 - 2x 3x &= 54 x &= 18end{align*}]Thus, there are 18 piles with 1 piece, leaving ( 27 - 18 = 9 ) piles with either 2 or 3 pieces. Let ( y ) be the number of piles with 2 pieces and ( z ) be the number of piles with 3 pieces. We have the following equations:[begin{align*}y + z &= 9 2y + 3z &= 40 - 18 = 22end{align*}]Solving these equations, we substitute ( z = 9 - y ) into the second equation:[begin{align*}2y + 3(9 - y) &= 22 2y + 27 - 3y &= 22 -y + 27 &= 22 -y &= -5 y &= 5end{align*}]Thus, the number of piles with exactly 2 chess pieces is ( y = 5 ).[boxed{5}],<think>Okay, let me try to figure out this problem step by step. So, we have to divide 40 chess pieces into 27 piles. Each pile can have 1, 2, or 3 pieces. The key information here is that the number of piles with only 1 chess piece is twice the number of the remaining piles. Then we need to find the number of piles with exactly 2 chess pieces.First, let me parse the problem again to make sure I understand it correctly. We have 27 piles total. Each pile has either 1, 2, or 3 pieces. The number of piles with 1 piece is twice the number of the remaining piles. So, the remaining piles would be those that have 2 or 3 pieces. Let me denote:Let’s define variables:Let x be the number of piles with 1 piece.Let y be the number of piles with 2 pieces.Let z be the number of piles with 3 pieces.Given that the total number of piles is 27, so:x + y + z = 27.The total number of chess pieces is 40, so:1*x + 2*y + 3*z = 40.The problem states that the number of piles with only 1 chess piece (which is x) is twice the number of the remaining piles. The remaining piles are those that are not with 1 piece, so that's y + z. So according to the problem:x = 2*(y + z).So now we have three equations:1. x + y + z = 272. x + 2y + 3z = 403. x = 2(y + z)Our goal is to find y, the number of piles with exactly 2 chess pieces.Let me substitute equation 3 into equations 1 and 2.From equation 3: x = 2(y + z). Let's plug that into equation 1:2(y + z) + y + z = 27Simplify:2y + 2z + y + z = 273y + 3z = 27Divide both sides by 3:y + z = 9.So, from this, we have y + z = 9. Let's keep this as equation 4.Now, plug x = 2(y + z) into equation 2:2(y + z) + 2y + 3z = 40But since y + z = 9 from equation 4, substitute that into the equation:2*9 + 2y + 3z = 4018 + 2y + 3z = 40Subtract 18 from both sides:2y + 3z = 22.Now, from equation 4, we have y + z = 9. So, we can write z = 9 - y. Substitute this into 2y + 3z = 22:2y + 3*(9 - y) = 222y + 27 - 3y = 22- y + 27 = 22Subtract 27 from both sides:- y = -5Multiply both sides by -1:y = 5.So, the number of piles with exactly 2 chess pieces is 5. Let me check if this makes sense.If y = 5, then z = 9 - y = 4. Then x = 2*(y + z) = 2*(9) = 18. So x = 18, y = 5, z = 4.Check total piles: 18 + 5 + 4 = 27. That's correct.Check total chess pieces: 18*1 + 5*2 + 4*3 = 18 + 10 + 12 = 40. That adds up.So, yes, the answer should be 5.But let me make sure I didn't misinterpret the problem. The problem says, "the number of piles with only 1 chess piece is twice the number of the remaining piles." So, the remaining piles are all the piles that aren't 1-piece piles. So, if there are x piles with 1 piece, then the remaining piles are 27 - x. But according to the problem, x = 2*(27 - x). Wait, hold on. Let me check that again.Wait, the problem says "the number of piles with only 1 chess piece is twice the number of the remaining piles." So, the remaining piles are those that are not 1-piece piles. So, if there are x piles with 1 piece, then remaining is 27 - x. So, the problem states that x = 2*(27 - x). Wait, that's different from what I did before. Wait, now I'm confused. Let me re-examine.Original interpretation: The number of piles with only 1 piece is twice the number of the remaining piles. So, "remaining piles" meaning the rest. So x = 2*(27 - x). Then solving that:x = 2*(27 - x)x = 54 - 2x3x = 54x = 18.Which gives x = 18, so remaining piles are 27 - 18 = 9. So that is consistent with the previous result where y + z = 9. So, my original setup was correct. Because in the problem statement, "the number of piles with only 1 chess piece is twice the number of the remaining piles." So, the remaining piles are the ones that aren't 1-piece piles. Therefore, x = 2*(remaining piles) = 2*(27 - x). Which leads to x = 18. Then remaining piles are 9, which are the y and z. Therefore, the equations are correct.So, the answer seems to check out. So, yes, y = 5. Therefore, the number of piles with exactly 2 chess pieces is 5.Final Answerboxed{5}</think>Given 40 chess pieces divided into 27 piles where each pile contains 1, 2, or 3 pieces, and the number of piles with 1 piece is twice the number of the remaining piles, we need to find the number of piles with exactly 2 pieces.Define variables:- ( x ) = number of piles with 1 piece- ( y ) = number of piles with 2 pieces- ( z ) = number of piles with 3 piecesWe have the following equations:1. ( x + y + z = 27 )2. ( x + 2y + 3z = 40 )3. ( x = 2(y + z) )Substitute ( x = 2(y + z) ) into equation 1:[2(y + z) + y + z = 27 implies 3(y + z) = 27 implies y + z = 9]Substitute ( x = 2(y + z) ) and ( y + z = 9 ) into equation 2:[2(9) + 2y + 3z = 40 implies 18 + 2y + 3z = 40 implies 2y + 3z = 22]Since ( y + z = 9 ), we substitute ( z = 9 - y ) into ( 2y + 3z = 22 ):[2y + 3(9 - y) = 22 implies 2y + 27 - 3y = 22 implies -y + 27 = 22 implies y = 5]Thus, the number of piles with exactly 2 chess pieces is ( boxed{5} ).]

🔑:Reference answer: 5

🧠:[<think>Okay, so I need to solve this problem where P is the sum of absolute values from |1 - 2x| up to |1 - 10x|. The question says that for any allowed value of x within a certain range, P is a constant. I need to find that constant and the options are 2, 3, 4, or 5. First, let me make sure I understand the problem correctly. P is the sum of absolute values: |1 - 2x| + |1 - 3x| + ... + |1 - 10x|. And the claim is that there's a specific range of x where, no matter what x you pick in that range, P remains the same constant value. My job is to figure out what that constant is. Hmm, absolute value functions are V-shaped, so each term |1 - nx| will have a minimum at x = 1/n. The sum of absolute values would typically have a piecewise linear shape with changes in slope at each critical point x = 1/n. However, if there's a range of x where the sum is constant, that means the slopes from each term must cancel out to zero in that interval. So, in other words, for x in that interval, the derivative of P with respect to x (if it exists) would be zero. But since absolute values are not differentiable at their corners, maybe we need to look at the intervals between those critical points and see if in one of those intervals the sum becomes flat.So first, let's list all the critical points where the expression inside any absolute value becomes zero. For each term |1 - nx|, the critical point is at x = 1/n. So for n from 2 to 10, the critical points are x = 1/2, 1/3, 1/4, ..., 1/10. These are the points where each individual absolute value term changes its slope from negative to positive or vice versa.Now, to find the interval where P is constant, I need to check if there exists an interval between two consecutive critical points where the sum of the slopes of all the terms is zero. Because if the total slope is zero, the function is constant there.So first, let's order the critical points from smallest to largest. The critical points are x = 1/10, 1/9, 1/8, ..., 1/2. So ordered from smallest to largest: 1/10 ≈ 0.1, 1/9 ≈ 0.111..., 1/8 = 0.125, 1/7 ≈ 0.142..., 1/6 ≈ 0.166..., 1/5 = 0.2, 1/4 = 0.25, 1/3 ≈ 0.333..., 1/2 = 0.5. So these are ordered as 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2.Now, between each pair of consecutive critical points, the sign of each term |1 - nx| will be determined. For x < 1/n, the term 1 - nx is positive, so |1 - nx| = 1 - nx. For x > 1/n, the term becomes negative, so |1 - nx| = nx - 1. Therefore, in each interval between critical points, each term has a constant slope: -n if x < 1/n, and +n if x > 1/n.Therefore, the total slope of P in any interval between two critical points is the sum of the slopes of each individual term. So in a particular interval, if x is between a and b, then for each n, if 1/n < a, then the slope is +n, and if 1/n > b, the slope is -n. Wait, actually, let's think carefully. Suppose we have an interval (c, d). For each n, if c > 1/n, then x > 1/n in this interval, so the term |1 - nx| has slope +n. If d < 1/n, then x < 1/n, so slope -n. If 1/n is within (c, d), then in that interval, the term |1 - nx| would have different slopes on either side of 1/n, but since we are considering intervals between critical points, 1/n should not be in the interval. Wait, but the intervals are between consecutive critical points, so each interval is between two consecutive 1/n and 1/m where n and m are such that 1/n < 1/m. Wait, actually, since the critical points are ordered from smallest to largest, each interval is (1/(k+1), 1/k) for k from 2 up? Wait, maybe not.Wait, let's take an example. The critical points are 1/10, 1/9, 1/8, ..., 1/2. So between 1/10 and 1/9, then between 1/9 and 1/8, etc., up to between 1/3 and 1/2. So each interval (1/(k+1), 1/k) for k from 2 to 9. Wait, but k here is the denominator. Wait, perhaps it's better to list them as intervals:1. (1/10, 1/9)2. (1/9, 1/8)3. (1/8, 1/7)4. (1/7, 1/6)5. (1/6, 1/5)6. (1/5, 1/4)7. (1/4, 1/3)8. (1/3, 1/2)So eight intervals. In each interval, all the terms |1 - nx| where n <= 1/x will have x >= 1/n, so those terms will have slope +n, and terms where n > 1/x will have x < 1/n, so slope -n.Wait, but in each interval, x is between two critical points, say between 1/(k+1) and 1/k. Then for each term |1 - nx|, n can be from 2 to 10. For each such n, if 1/n < x, then slope is +n, else slope is -n. So in the interval (1/(k+1), 1/k), the numbers n where 1/n < x are n < 1/x. But x is between 1/(k+1) and 1/k, so 1/x is between k and k+1. Therefore, n < 1/x corresponds to n <= k. Therefore, in the interval (1/(k+1), 1/k), the slope contributions are +n for n <= k and -n for n > k. Therefore, total slope S in that interval would be sum_{n=2}^k n - sum_{n=k+1}^{10} n.Therefore, S = [sum from n=2 to k of n] - [sum from n=k+1 to 10 of n]. Then, if S = 0, the function is constant over that interval. So we need to find k such that sum_{n=2}^k n = sum_{n=k+1}^{10} n.So the problem reduces to finding k between 2 and 9 (since k+1 has to be at most 10) such that the sum from 2 to k equals the sum from k+1 to 10.Let me compute the total sum first. Sum from n=2 to 10 is (10*11)/2 - 1 = 55 - 1 = 54. Therefore, if sum from 2 to k is equal to sum from k+1 to 10, then each sum must be 54/2 = 27. So we need to find k such that sum from 2 to k is 27. Let's compute cumulative sums:Sum from 2 to k:k=2: 2k=3: 2+3=5k=4: 5+4=9k=5: 9+5=14k=6: 14+6=20k=7: 20+7=27Ah! At k=7, the sum from 2 to 7 is 27. Therefore, when k=7, the sum from 2 to 7 is 27, and the sum from 8 to 10 is 8+9+10=27. Therefore, for k=7, the total slope S would be 27 - 27 = 0.Therefore, in the interval (1/(k+1), 1/k) where k=7, that is, (1/8, 1/7), the total slope is zero. Therefore, in that interval, the function P is constant. Wait, but 1/(k+1) when k=7 is 1/8, and 1/k is 1/7 ≈ 0.142. So the interval (1/8, 1/7) ≈ (0.125, 0.142). Therefore, if x is between 1/8 and 1/7, P is a constant. So the answer would be the value of P in this interval.So now, to compute the value of P in that interval. Let's pick an x in (1/8, 1/7), say x = 1/7.5 ≈ 0.133..., but to make calculation easier, maybe x=2/15 ≈0.133..., which is between 1/8=0.125 and 1/7≈0.142. Wait, 2/15 is approximately 0.1333. Let's check if 2/15 is between 1/8 and 1/7. 1/8=0.125, 2/15≈0.133, 1/7≈0.142. Yes, it is. Alternatively, maybe x=1/7.5, which is 2/15. Let's use x=2/15.But maybe it's easier to compute it in general. Since in the interval (1/8, 1/7), each term |1 - nx| can be expressed as:For n=2 to 7: since x >1/8, n=2: 1/8 ≈0.125, x >0.125, so for n=2, 1/n=0.5. Since x <1/7≈0.142, which is still less than 0.5. Wait, hold on. Wait, for n=2 to 7:Wait, for each n from 2 to 10, check if x >1/n or x <1/n. Since x is between 1/8 and 1/7, which is approximately 0.125 to 0.142. So 1/n for n=2 is 0.5, which is greater than x. So x <1/n for n=2 to 7? Wait, wait, no:Wait, n=2: 1/n=0.5, x≈0.133 <0.5, so |1 -2x|=1-2xn=3:1/3≈0.333, x≈0.133 <0.333, so |1 -3x|=1-3xn=4:1/4=0.25, x≈0.133 <0.25, so |1 -4x|=1-4xn=5:1/5=0.2, x≈0.133 <0.2, so |1 -5x|=1-5xn=6:1/6≈0.166, x≈0.133 <0.166, so |1 -6x|=1-6xn=7:1/7≈0.142, x≈0.133 <0.142, so |1 -7x|=1-7xn=8:1/8=0.125, x≈0.133 >0.125, so |1 -8x|=8x -1n=9:1/9≈0.111, x≈0.133 >0.111, so |1 -9x|=9x -1n=10:1/10=0.1, x≈0.133 >0.1, so |1 -10x|=10x -1Therefore, in this interval (1/8,1/7), for n=2 to7, |1 -nx|=1 -nx, and for n=8,9,10, |1 -nx|=nx -1. So, P can be written as sum_{n=2}^7 (1 - nx) + sum_{n=8}^{10} (nx -1).Let's compute this sum.First, compute sum_{n=2}^7 (1 - nx):This is equal to sum_{n=2}^7 1 - sum_{n=2}^7 nxNumber of terms from n=2 to7: 6 terms. So sum_{n=2}^7 1 =6*1=6Sum_{n=2}^7 nx= x*(2 +3 +4 +5 +6 +7)=x*(2+3+4+5+6+7)=x*(27)So sum_{n=2}^7 (1 - nx)=6 -27xSimilarly, sum_{n=8}^{10} (nx -1):This is sum_{n=8}^{10} nx - sum_{n=8}^{10} 1Number of terms from n=8 to10: 3 terms. So sum_{n=8}^{10}1=3*1=3Sum_{n=8}^{10}nx =x*(8 +9 +10)=x*27Therefore, sum_{n=8}^{10}(nx -1)=27x -3Thus, total P = (6 -27x) + (27x -3) =6 -27x +27x -3= (6 -3)+( -27x +27x)=3 +0=3.So P=3 in the interval (1/8,1/7). Therefore, the constant value is 3, which is option B.But let me verify this with a specific x in that interval. Let's take x=1/8.5, which is approximately 0.1176? Wait, no, 1/8 is 0.125, 1/7≈0.142. Let's take x=0.13, which is within (0.125,0.142).Compute each term:|1 -2*0.13| = |1 -0.26| =0.74|1 -3*0.13| = |1 -0.39| =0.61|1 -4*0.13| = |1 -0.52| =0.48|1 -5*0.13| = |1 -0.65| =0.35|1 -6*0.13| = |1 -0.78| =0.22|1 -7*0.13| = |1 -0.91| =0.09|1 -8*0.13| = |1 -1.04| =0.04Wait, but 8*0.13=1.04, so |1 -1.04|=0.04Wait, but that's positive? Wait, no. Wait, 8x=1.04, so 1 -8x= -0.04, absolute value is 0.04.Similarly, 9x=1.17, so |1 -9x|=| -0.17|=0.1710x=1.3, |1 -10x|=| -0.3|=0.3Now, adding all these up:0.74 +0.61 +0.48 +0.35 +0.22 +0.09 +0.04 +0.17 +0.3Let me compute step by step:0.74 +0.61 =1.351.35 +0.48=1.831.83 +0.35=2.182.18 +0.22=2.402.40 +0.09=2.492.49 +0.04=2.532.53 +0.17=2.702.70 +0.3=3.00Wow, that adds up exactly to 3. So at x=0.13, which is in (1/8,1/7), P=3. Let's check another x in that interval, say x=0.14.Compute each term:|1 -2*0.14|=|1 -0.28|=0.72|1 -3*0.14|=|1 -0.42|=0.58|1 -4*0.14|=|1 -0.56|=0.44|1 -5*0.14|=|1 -0.70|=0.30|1 -6*0.14|=|1 -0.84|=0.16|1 -7*0.14|=|1 -0.98|=0.02|1 -8*0.14|=|1 -1.12|=0.12|1 -9*0.14|=|1 -1.26|=0.26|1 -10*0.14|=|1 -1.40|=0.40Now, add them up:0.72 +0.58 =1.301.30 +0.44=1.741.74 +0.30=2.042.04 +0.16=2.202.20 +0.02=2.222.22 +0.12=2.342.34 +0.26=2.602.60 +0.40=3.00Again, it adds up to 3. So this confirms that in that interval, regardless of x, the sum is 3. Therefore, the constant is 3, which is option B.But just to be thorough, let me check with another x, like x=1/8. Let's try x=1/8=0.125, which is the lower bound of the interval. Wait, but 1/8 is a critical point. At x=1/8, the term |1 -8x|=|1 -1|=0. So let's compute P at x=1/8.Compute each term:|1 -2*(1/8)| = |1 -0.25|=0.75|1 -3*(1/8)| = |1 -0.375|=0.625|1 -4*(1/8)| = |1 -0.5|=0.5|1 -5*(1/8)| = |1 -0.625|=0.375|1 -6*(1/8)| = |1 -0.75|=0.25|1 -7*(1/8)| = |1 -0.875|=0.125|1 -8*(1/8)| = |1 -1|=0|1 -9*(1/8)| = |1 -1.125|=0.125|1 -10*(1/8)| = |1 -1.25|=0.25Adding these up:0.75 +0.625=1.3751.375 +0.5=1.8751.875 +0.375=2.252.25 +0.25=2.52.5 +0.125=2.6252.625 +0=2.6252.625 +0.125=2.752.75 +0.25=3.0So even at x=1/8, it's 3.0. Similarly, if I check x=1/7≈0.142, upper bound:Compute each term:|1 -2*(1/7)| ≈ |1 -0.2857|≈0.7143|1 -3*(1/7)|≈|1 -0.4286|≈0.5714|1 -4*(1/7)|≈|1 -0.5714|≈0.4286|1 -5*(1/7)|≈|1 -0.7143|≈0.2857|1 -6*(1/7)|≈|1 -0.8571|≈0.1429|1 -7*(1/7)|= |1 -1|=0|1 -8*(1/7)|≈|1 -1.1429|≈0.1429|1 -9*(1/7)|≈|1 -1.2857|≈0.2857|1 -10*(1/7)|≈|1 -1.4286|≈0.4286Adding them:0.7143 +0.5714 ≈1.28571.2857 +0.4286≈1.71431.7143 +0.2857≈2.02.0 +0.1429≈2.14292.1429 +0≈2.14292.1429 +0.1429≈2.28582.2858 +0.2857≈2.57152.5715 +0.4286≈3.0001Which is approximately 3. So due to rounding errors, it's about 3. Therefore, even at the critical points, the value is 3. However, at the critical points, the function might not be differentiable, but the value itself is still 3. So the interval where P is constant includes the endpoints? Wait, but actually, when we have absolute value functions, at the critical points, the function is continuous but the derivative changes. However, if the interval is defined as closed interval [1/8,1/7], then P is constant throughout. But depending on the problem statement, "within a certain specific range", so maybe the allowed range is [1/8,1/7], and for any x in that range, P is constant. Therefore, the answer is 3, which is option B.Therefore, I think the correct answer is B) 3.Final Answerboxed{B}</think>Given the problem where ( P = |1 - 2x| + |1 - 3x| + |1 - 4x| + cdots + |1 - 9x| + |1 - 10x| ) is a constant for any allowed value of ( x ) within a certain range, we need to determine this constant.1. Critical Points Identification: The critical points where each term ( |1 - nx| ) changes its slope are ( x = frac{1}{2}, frac{1}{3}, frac{1}{4}, ldots, frac{1}{10} ). These points are ordered from smallest to largest: ( frac{1}{10}, frac{1}{9}, frac{1}{8}, ldots, frac{1}{2} ).2. Interval Analysis: We need to find an interval between two consecutive critical points where the sum of the slopes of all terms is zero. This interval will be where ( P ) is constant. The key is to find ( k ) such that the sum of slopes from ( n = 2 ) to ( k ) equals the sum of slopes from ( n = k+1 ) to ( 10 ).3. Sum Calculation: The total sum of the terms from ( n = 2 ) to ( 10 ) is 54. We need this sum to be split equally (27 each) for the slopes to cancel out. The sum from ( n = 2 ) to ( 7 ) is 27, and the sum from ( n = 8 ) to ( 10 ) is also 27. This occurs at ( k = 7 ).4. Interval Verification: The interval corresponding to ( k = 7 ) is ( left( frac{1}{8}, frac{1}{7} right) ). Within this interval, the terms ( |1 - nx| ) for ( n = 2 ) to ( 7 ) are positive, and for ( n = 8 ) to ( 10 ) are negative. The sum of these terms is calculated to be constant.5. Value Calculation: For ( x ) in ( left( frac{1}{8}, frac{1}{7} right) ), the expression simplifies to: [ P = sum_{n=2}^{7} (1 - nx) + sum_{n=8}^{10} (nx - 1) ] This simplifies to ( 6 - 27x + 27x - 3 = 3 ).6. Verification with Specific Values: Checking specific values within the interval confirms that ( P = 3 ).Thus, the constant value of ( P ) is (boxed{B}).,<think>Okay, so I need to figure out the constant value of P when it doesn't change for any allowed x within a certain range. The expression given is P = |1 - 2x| + |1 - 3x| + |1 - 4x| + ... + |1 - 9x| + |1 - 10x|. The answer choices are 2, 3, 4, or 5. First, I remember that the sum of absolute values like this can sometimes be constant if all the absolute value terms are "balanced" in such a way that their increases and decreases cancel out. But how does that happen? Maybe there's a specific range of x where each term inside the absolute value is either always non-negative or always non-positive, but since it's supposed to be a constant regardless of x in that range, the combination of these terms must somehow add up to the same total.Wait, but if x is in a range where each term |1 - nx| doesn't change its sign (either always positive or always negative), then the absolute value can be removed, and the expression becomes linear in x. But if P is supposed to be a constant, then the coefficients of x in that linear expression must sum to zero. That is, the sum of the derivatives (slopes) of each term must cancel out. Let me try to formalize this.Let's denote each term as |1 - nx| where n ranges from 2 to 10. For each n, if 1 - nx is positive, then |1 - nx| = 1 - nx; if 1 - nx is negative, then |1 - nx| = nx - 1. The derivative (slope) of each term with respect to x is -n if 1 - nx > 0 and +n if 1 - nx < 0. If P is constant, then the total derivative (sum of all individual slopes) must be zero. So, if we can find an x such that the sum of the slopes of all terms is zero, and in some interval around that x, the individual terms don't change their sign (i.e., the critical points where 1 - nx = 0 are outside the interval), then P would be constant in that interval.So, first, let's find the critical points for each term. For each n, the critical point is x = 1/n. So, the critical points are at x = 1/2, 1/3, 1/4, ..., 1/10. These are the points where each individual absolute value term changes its slope. Now, the problem states that there is a certain specific range of x where P is constant. That range must be between two consecutive critical points where the total slope doesn't change sign. Wait, no, actually, within an interval between two consecutive critical points, each term's slope is fixed (either positive or negative), so the total slope is the sum of all these fixed slopes. For the total expression P to be constant, the total slope in that interval must be zero.Therefore, we need to find an interval (a, b) where for all x in (a, b), the sign of each 1 - nx is fixed (i.e., either all terms to the left of their critical points or all to the right), such that when we add up all the slopes (either +n or -n), the total sum is zero. Then, in that interval, P would be a linear function with slope zero, hence constant.So, let's list all critical points from n=2 to n=10:n=2: x=1/2=0.5n=3: x=1/3≈0.333...n=4: x=1/4=0.25n=5: x=1/5=0.2n=6: x=1/6≈0.166...n=7: x=1/7≈0.142...n=8: x=1/8=0.125n=9: x=1/9≈0.111...n=10: x=1/10=0.1So the critical points in order from smallest to largest are:1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2.Therefore, the intervals between critical points are:(0, 1/10), (1/10, 1/9), (1/9, 1/8), ..., up to (1/3, 1/2), and then (1/2, ∞). But since the problem states "any allowed value of x within a certain specific range", we need to check in which interval the total slope is zero.In each interval between two consecutive critical points, the terms corresponding to n ≤ 1/x will have 1 - nx ≥ 0 (since x < 1/n), so their slopes will be -n. The terms with n > 1/x will have 1 - nx < 0, so their slopes will be +n. Wait, actually, when x increases past 1/n, the term |1 - nx| switches from decreasing (slope -n) to increasing (slope +n). So, for x < 1/n: slope is -n, for x > 1/n: slope is +n.Therefore, in any interval between two critical points x ∈ (1/(k+1), 1/k), the terms with n ≤ k will have slope +n (since x > 1/n), and terms with n > k will have slope -n (since x < 1/n). Wait, no, if x is in (1/(k+1), 1/k), then for n ≤ k, 1/n ≥ 1/k, so x < 1/n (since x < 1/k). Wait, that's confusing. Let's take an example.Suppose x is between 1/5 and 1/4 (0.2 and 0.25). Then, for n=2, 1/n=0.5, x < 0.5, so |1 - 2x| is 1 - 2x, slope -2.Similarly for n=3: 1/3 ≈0.333, x < 0.333, so |1 - 3x|=1 - 3x, slope -3.n=4: 1/4=0.25, x is in (0.2,0.25), so x < 1/4? No, x is between 0.2 and 0.25. So for n=4, when x is in (0.2, 0.25), x is less than 1/4 (0.25), so |1 - 4x| = 1 - 4x, slope -4.Wait, but if x is approaching 0.25 from below, say x=0.24, then n=4: 1 - 4*0.24=1 - 0.96=0.04>0, so slope is -4.But for n=5: 1/n=0.2, x is in (0.2,0.25), so x > 0.2. Therefore, |1 - 5x| = 5x -1, slope +5.Similarly, n=6: x=0.2 to 0.25 is greater than 1/6≈0.166, so |1 -6x|=6x -1, slope +6.Same for n=7,8,9,10: x=0.2 to 0.25 is greater than 1/7≈0.142, etc., so all those terms will have slopes +7, +8, +9, +10.Therefore, in the interval (1/5,1/4)≈(0.2,0.25), the slopes are:For n=2: slope -2n=3: slope -3n=4: slope -4n=5: slope +5n=6: slope +6n=7: slope +7n=8: slope +8n=9: slope +9n=10: slope +10Therefore, total slope is (-2 -3 -4) + (5+6+7+8+9+10) = (-9) + (45) = 36. So total slope is 36 in this interval, which is positive, so P is increasing here. Thus, P is not constant here.Hmm, so to have P be constant, the total slope must be zero. So we need to find an interval where the sum of positive slopes equals the sum of negative slopes. Let's try to see if such an interval exists.Let me think. Let's denote for a given interval between 1/(k+1) and 1/k, the terms with n ≤ k will be in the region x < 1/n, so their slopes are negative (-n). Wait, no: Wait, when x is in the interval (1/(k+1), 1/k), then for each term n:If n ≤ k: Since 1/n ≥ 1/k, and x is less than 1/k, so x < 1/n, so |1 - nx| = 1 - nx, slope -n.If n ≥ k+1: Since 1/n ≤ 1/(k+1), and x is greater than 1/(k+1), so x > 1/n, so |1 - nx| = nx -1, slope +n.Therefore, total slope S in the interval (1/(k+1),1/k) is:S = [sum_{n=2}^k (-n)] + [sum_{n=k+1}^{10} (+n)]So S = -sum_{n=2}^k n + sum_{n=k+1}^{10} nWe need S = 0.So, sum_{n=k+1}^{10} n = sum_{n=2}^k nWhich means sum_{n=2}^{10} n = 2 * sum_{n=2}^k nBut sum from n=2 to 10 is (10*11)/2 -1 = 55 -1=54.Wait, sum from n=2 to 10 is 54. So if sum_{n=k+1}^{10} n = sum_{n=2}^k n, then each sum is 54/2=27.Therefore, we need sum from n=2 to k equal 27, and sum from n=k+1 to 10 also equal 27.So, the total sum is 54, so each half is 27.Therefore, we need to find k such that sum_{n=2}^k n =27.Let's compute cumulative sums:sum from n=2 to k:k=2: 2k=3: 2+3=5k=4: 5+4=9k=5: 9+5=14k=6:14+6=20k=7:20+7=27Ah, here we go. At k=7, the sum from n=2 to 7 is 2+3+4+5+6+7=27. Then the sum from n=8 to 10 is 8+9+10=27. Perfect, so when k=7, the sum from 2 to 7 is 27, and from 8 to 10 is also 27. Therefore, in the interval (1/(k+1), 1/k) = (1/8,1/7)≈(0.125,0.1428...), the total slope S would be zero. Therefore, in that interval, P is constant.Therefore, the answer should be the value of P in that interval. Let's compute P in that interval.Since x is between 1/8 and 1/7, so for n=2 to 7, since x <1/n (because 1/n for n=7 is ≈0.1428, and x <1/7≈0.1428 but the interval is (1/8,1/7), so x is less than 1/7 but greater than 1/8≈0.125. Wait, for n=7, x is less than 1/7 (since x <1/7), so |1 -7x|=1 -7x. Similarly, for n=8 to 10, x >1/n (since x >1/8=0.125, n=8: 1/8=0.125, so x>0.125; but the interval is (1/8,1/7), so x is greater than 1/8 but less than 1/7. So for n=8, x >1/8, so |1 -8x|=8x -1. Similarly for n=9 and 10.Therefore, in this interval:For n=2 to7: |1 -nx|=1 -nxFor n=8 to10: |1 -nx|=nx -1Therefore, P = sum_{n=2}^7 (1 -nx) + sum_{n=8}^{10} (nx -1)Compute sum_{n=2}^7 (1 -nx) = sum_{n=2}^7 1 - sum_{n=2}^7 nx = (7-1)*1 - x sum_{n=2}^7 nWait, sum from n=2 to7 of 1 is 6*1=6.Sum from n=2 to7 of n is 2+3+4+5+6+7=27.Therefore, sum_{n=2}^7 (1 -nx) = 6 -27x.Similarly, sum_{n=8}^{10} (nx -1) = sum_{n=8}^{10} nx - sum_{n=8}^{10}1 = x sum_{n=8}^{10}n - 3*1Sum from n=8 to10 of n is 8+9+10=27.Therefore, sum_{n=8}^{10} (nx -1) =27x -3.Therefore, total P = (6 -27x) + (27x -3) =6 -3=3.So P=3. Therefore, the constant value is 3, which is option (B).Let me verify that. So, when x is between 1/8 and1/7, P=3. Let's check with a specific value in that interval, say x=1/7.5≈0.1333 (which is between 1/8=0.125 and1/7≈0.1428).Compute each term:For n=2: |1 -2*(1/7.5)| = |1 - 2/7.5| = |1 - 0.2666...| = 0.7333...n=3: |1 -3/7.5| = |1 -0.4| =0.6n=4: |1 -4/7.5| = |1 -0.5333...| =0.4666...n=5: |1 -5/7.5| = |1 -0.6666...| =0.3333...n=6: |1 -6/7.5| = |1 -0.8| =0.2n=7: |1 -7/7.5| = |1 -0.9333...| =0.0666...n=8: |1 -8/7.5| = |1 -1.0666...| =0.0666...n=9: |1 -9/7.5| = |1 -1.2| =0.2n=10: |1 -10/7.5| = |1 -1.3333...| =0.3333...Now sum them up:0.7333 +0.6 +0.4666 +0.3333 +0.2 +0.0666 +0.0666 +0.2 +0.3333Let's compute step by step:Start with 0.7333+0.6 =1.3333+0.4666=1.8+0.3333=2.1333+0.2=2.3333+0.0666=2.4+0.0666=2.4666+0.2=2.6666+0.3333=3.0Yes! The total is 3. So that checks out. Therefore, the constant value is 3, which is option B.Final Answerboxed{B}</think>To determine the constant value of ( P = |1 - 2x| + |1 - 3x| + |1 - 4x| + cdots + |1 - 9x| + |1 - 10x| ) within a certain range of ( x ), we need to find an interval where the total slope of the absolute value terms sums to zero. The critical points where each term ( |1 - nx| ) changes its slope are at ( x = frac{1}{n} ) for ( n ) from 2 to 10. These critical points are ordered as ( frac{1}{10}, frac{1}{9}, frac{1}{8}, frac{1}{7}, frac{1}{6}, frac{1}{5}, frac{1}{4}, frac{1}{3}, frac{1}{2} ).We need to find an interval where the sum of the slopes of the terms is zero. For an interval ( left( frac{1}{k+1}, frac{1}{k} right) ), the slopes of the terms ( |1 - nx| ) for ( n leq k ) are negative, and for ( n geq k+1 ) are positive. The total slope is zero if the sum of the positive slopes equals the sum of the negative slopes.Calculating the sums, we find that for ( k = 7 ), the sum of the terms from ( n = 2 ) to ( 7 ) is 27, and the sum of the terms from ( n = 8 ) to ( 10 ) is also 27. This balance occurs in the interval ( left( frac{1}{8}, frac{1}{7} right) ).Within this interval, the terms ( |1 - nx| ) for ( n = 2 ) to ( 7 ) are ( 1 - nx ), and for ( n = 8 ) to ( 10 ) are ( nx - 1 ). Summing these terms:[begin{aligned}sum_{n=2}^{7} (1 - nx) + sum_{n=8}^{10} (nx - 1) &= left( 6 - 27x right) + left( 27x - 3 right) &= 6 - 3 &= 3.end{aligned}]Thus, the constant value of ( P ) is ( 3 ).[boxed{B}]]

🔑:B

🧠:[<think>Okay, let me try to figure out this problem. So, the equation given is 2010^{2011} minus 2010^{2009} equals 2010^x multiplied by 2009 and 2011. I need to find the value of x from the options given. Hmm, let's break this down step by step.First, looking at the left side of the equation: 2010^{2011} - 2010^{2009}. Both terms have the same base, which is 2010, but different exponents. Maybe I can factor out a common term here. The exponents are 2011 and 2009, so the smaller one is 2009. If I factor out 2010^{2009}, what's left?Let me write that out: 2010^{2009}(2010^{2} - 1). Wait, because 2010^{2011} is 2010^{2009 + 2} which is 2010^{2009} * 2010^2. Similarly, the second term is just 2010^{2009}, so when I factor that out, it's like 2010^{2009}(2010^2 - 1). That makes sense.So, the left side simplifies to 2010^{2009}*(2010^2 - 1). Now, the right side of the equation is 2010^x * 2009 * 2011. So, we have:2010^{2009}*(2010^2 - 1) = 2010^x * 2009 * 2011.Hmm, okay. Let me see. The term (2010^2 - 1) can be factored further since it's a difference of squares. Remember, a^2 - b^2 = (a - b)(a + b). Here, a is 2010 and b is 1. So, 2010^2 - 1 = (2010 - 1)(2010 + 1) = 2009 * 2011. Wait, that's perfect because on the right side, we have 2009 * 2011. So substituting that in, the left side becomes:2010^{2009} * 2009 * 2011.Therefore, the equation simplifies to:2010^{2009} * 2009 * 2011 = 2010^x * 2009 * 2011.Now, if I divide both sides by 2009 * 2011, those terms cancel out, and we get:2010^{2009} = 2010^x.Since the bases are the same and the equation holds, the exponents must be equal. Therefore, x must be 2009. Let me check the options again. The options are A:2008, B:2009, C:2010, D:2011. So, the answer should be B:2009.Wait a second, let me verify that once more to make sure I didn't skip any steps. Starting from the original equation:2010^{2011} - 2010^{2009} = 2010^x * 2009 * 2011.Factoring the left side: 2010^{2009}(2010^2 - 1) = 2010^x * 2009 * 2011.Then, factoring 2010^2 - 1 as (2010 -1)(2010 +1) = 2009*2011. So replacing that in, we get:2010^{2009} *2009 *2011 = 2010^x *2009*2011.Cancel out 2009*2011 from both sides, which leaves 2010^{2009} = 2010^x, so x=2009. Yep, that seems right. So the answer is B.But just to be thorough, let me make sure there's no alternative way someone might approach this where they could make a mistake. For example, maybe someone might think to subtract the exponents first, but exponents don't work like that when subtracting terms. It's important to factor out the common term instead. So, for example, 2010^{2011} - 2010^{2009} can't be simplified by subtracting exponents, but factoring out 2010^{2009} is the right approach. Then, recognizing that 2010^2 -1 factors into 2009*2011 is key here because that's what allows the 2009*2011 terms to cancel out on both sides. So, that step is crucial. If someone didn't factor the difference of squares, they might get stuck. But once you factor that, everything lines up, and the exponents have to be equal. So, yeah, definitely x is 2009. Another check: suppose x was 2008, then the right side would be 2010^{2008}*2009*2011. But the left side is 2010^{2009}*2009*2011. Since 2010^{2009} is 2010 times 2010^{2008}, so the left side is 2010 times larger than the right side if x was 2008, which wouldn't make sense. Similarly, if x was 2010, then the right side would be 2010^{2010}*2009*2011, which is 2010 times larger than the left side, so that's not possible. If x was 2011, it's even larger. So, only when x is 2009 do both sides match. Therefore, the answer is B:2009. I feel confident about this now.Final Answerboxed{B}</think>Given the equation (2010^{2011} - 2010^{2009} = 2010^x times 2009 times 2011), we need to find the value of (x).First, we factor out the common term (2010^{2009}) from the left side:[2010^{2011} - 2010^{2009} = 2010^{2009}(2010^2 - 1)]Next, we recognize that (2010^2 - 1) is a difference of squares, which can be factored as:[2010^2 - 1 = (2010 - 1)(2010 + 1) = 2009 times 2011]Substituting this back into the equation, we get:[2010^{2009} times 2009 times 2011 = 2010^x times 2009 times 2011]By dividing both sides of the equation by (2009 times 2011), we obtain:[2010^{2009} = 2010^x]Since the bases are the same, the exponents must be equal. Therefore, we find:[x = 2009]The value of (x) is (boxed{B}).,<think>Alright, let's tackle this problem step by step. So, we have the equation:2010^{2011} - 2010^{2009} = 2010^x × 2009 × 2011And we need to find the value of x from the given options. Hmm, okay. Let me think. First, looking at the left side of the equation, there are two terms: 2010 raised to the power of 2011 and 2010 raised to the power of 2009. Both terms have the same base, which is 2010. The exponents are different, but they are two apart—2011 and 2009. The right side of the equation is 2010 raised to some power x, multiplied by 2009 and 2011. Since the left side is a subtraction of two exponentials with the same base, maybe we can factor out the smaller exponent? Let me try that. So, 2010^{2011} minus 2010^{2009}... If I factor out 2010^{2009}, that should leave me with 2010^{2011 - 2009} minus 1, right? Because when you factor out a term, you subtract the exponents. Let me verify that. If I factor 2010^{2009} out of both terms, the first term becomes 2010^{2009} × 2010^{2} (since 2011 - 2009 = 2), and the second term is just 2010^{2009} × 1. So, yes, factoring out 2010^{2009} gives:2010^{2009} × (2010^2 - 1) = 2010^x × 2009 × 2011Okay, so that's the left side factored. Now, the equation becomes:2010^{2009} × (2010^2 - 1) = 2010^x × 2009 × 2011Now, let's look at the term (2010^2 - 1). That looks like a difference of squares, right? Because a^2 - b^2 = (a - b)(a + b). Here, a is 2010 and b is 1. So, 2010^2 - 1 = (2010 - 1)(2010 + 1) = 2009 × 2011. Oh, that's convenient because on the right side we have 2009 × 2011 as well. So substituting back in, the left side becomes:2010^{2009} × 2009 × 2011 = 2010^x × 2009 × 2011Now, both sides of the equation have 2009 × 2011. Assuming that 2009 and 2011 are not zero (which they aren't; they're both positive integers), we can divide both sides by 2009 × 2011 to simplify the equation. Let's do that:2010^{2009} = 2010^xNow, since the bases are the same and the equation holds, the exponents must be equal. Therefore, x must be 2009. Wait, but let me check again to make sure I didn't make a mistake. Let's go through each step:1. Original equation: 2010^{2011} - 2010^{2009} = 2010^x × 2009 × 20112. Factor left side: 2010^{2009}(2010^2 - 1) = 2010^x × 2009 × 20113. Recognize that 2010^2 - 1 is (2010 - 1)(2010 + 1) = 2009 × 20114. Substitute back in: 2010^{2009} × 2009 × 2011 = 2010^x × 2009 × 20115. Cancel out 2009 × 2011 from both sides: 2010^{2009} = 2010^x6. Therefore, x = 2009That seems correct. So the answer should be B: 2009. Let me check the options again. The options are A: 2008, B: 2009, C: 2010, D: 2011. Yes, B is 2009. But wait, just to be thorough, let me test if any other answer makes sense. Suppose x was 2010, then the right side would be 2010^{2010} × 2009 × 2011. Comparing that to the left side after factoring: 2010^{2009} × 2009 × 2011. So the left side is 2010^{2009} and the right side would be 2010^{2010}, which is 2010 times larger. So that's not equal. Similarly, if x was 2008, then the right side would be 2010^{2008} × 2009 × 2011, which is 2010 times smaller than the left side. So that doesn't work either. And if x was 2011, the right side would be 2010^{2011} × 2009 × 2011, which is way larger than the left side. So none of the other options work except for x=2009. Therefore, I'm confident that the correct answer is B: 2009.Final Answerboxed{B}</think>Given the equation:[ 2010^{2011} - 2010^{2009} = 2010^x times 2009 times 2011 ]We start by factoring the left-hand side. We factor out the common term (2010^{2009}):[ 2010^{2009} times (2010^2 - 1) ]Next, we recognize that (2010^2 - 1) is a difference of squares, which can be factored as:[ (2010 - 1)(2010 + 1) = 2009 times 2011 ]Substituting this back into the equation, we get:[ 2010^{2009} times 2009 times 2011 = 2010^x times 2009 times 2011 ]We can divide both sides of the equation by (2009 times 2011):[ 2010^{2009} = 2010^x ]Since the bases are the same, the exponents must be equal:[ x = 2009 ]Thus, the value of (x) is (boxed{B}).]

🔑:Solution: We have 2010^{2011} - 2010^{2009} = 2010^{2009}(2010^2 - 1) = 2010^{2009}(2010 - 1)(2010 + 1) = 2010^{2009} times 2009 times 2011.Given 2010^{2011} - 2010^{2009} = 2010^x times 2009 times 2011,we have 2010^{2009} times 2009 times 2011 = 2010^x times 2009 times 2011, thus x = 2009.Therefore, the correct answer is boxed{text{B}}.