🧠:[<think>Okay, so I need to find the value of k such that the sum of the squares of the distances from point P to points A and B on the ellipse only depends on the slope k and not on the position of P on the major axis. Let me try to break this down step by step.First, let me recall the equation of the given ellipse: x²/25 + y²/16 = 1. So, the semi-major axis is 5 (since 25 = 5²) along the x-axis, and the semi-minor axis is 4 (16 = 4²) along the y-axis. Therefore, the major axis is the x-axis from (-5, 0) to (5, 0).Point P is a moving point on the major axis, so its coordinates can be represented as (h, 0), where h is between -5 and 5. Now, a line with slope k passes through P and intersects the ellipse at points A and B. We need to find k such that |PA|² + |PB|² is independent of h, meaning it's only a function of k and not h.To approach this, I'll probably need to parametrize the line passing through P with slope k and find its intersection points with the ellipse. Then, calculate the distances PA and PB, square them, add them up, and then set the expression such that all terms involving h cancel out, leaving an expression that only depends on k. The value of k that allows this cancellation is the answer.Let me start by writing the equation of the line passing through P(h, 0) with slope k. The general equation is y = k(x - h). So, y = kx - kh.Now, we need to find the points where this line intersects the ellipse. Substitute y = kx - kh into the ellipse equation:x²/25 + (kx - kh)²/16 = 1.Expanding the second term:(kx - kh)² = k²x² - 2k²h x + k²h².So, substituting back into the ellipse equation:x²/25 + [k²x² - 2k²h x + k²h²]/16 = 1.Multiply through by 400 (the least common multiple of 25 and 16) to eliminate denominators:16x² + 25(k²x² - 2k²h x + k²h²) = 400.Expanding this:16x² + 25k²x² - 50k²h x + 25k²h² = 400.Combine like terms:(16 + 25k²)x² - 50k²h x + (25k²h² - 400) = 0.This is a quadratic equation in x. Let's denote the quadratic equation as Ax² + Bx + C = 0, where:A = 16 + 25k²,B = -50k²h,C = 25k²h² - 400.Let x₁ and x₂ be the roots of this equation, which correspond to the x-coordinates of points A and B. Then, according to Vieta's formulas:x₁ + x₂ = -B/A = (50k²h)/(16 + 25k²),x₁x₂ = C/A = (25k²h² - 400)/(16 + 25k²).Now, the points A and B lie on the line y = k(x - h), so their coordinates are (x₁, k(x₁ - h)) and (x₂, k(x₂ - h)), respectively.Point P is (h, 0). Therefore, the distances PA and PB can be calculated using the distance formula.Distance PA squared:|PA|² = (x₁ - h)² + [k(x₁ - h) - 0]^2= (x₁ - h)^2 + k²(x₁ - h)^2= (1 + k²)(x₁ - h)^2.Similarly, |PB|² = (1 + k²)(x₂ - h)^2.Therefore, |PA|² + |PB|² = (1 + k²)[(x₁ - h)^2 + (x₂ - h)^2].Now, let's expand the terms inside the brackets:(x₁ - h)^2 + (x₂ - h)^2 = (x₁^2 - 2hx₁ + h²) + (x₂^2 - 2hx₂ + h²)= x₁² + x₂² - 2h(x₁ + x₂) + 2h².We can express x₁² + x₂² using Vieta's formulas. Recall that x₁² + x₂² = (x₁ + x₂)^2 - 2x₁x₂.Therefore, substituting back:(x₁ - h)^2 + (x₂ - h)^2 = (x₁ + x₂)^2 - 2x₁x₂ - 2h(x₁ + x₂) + 2h².Now, substituting the expressions for x₁ + x₂ and x₁x₂ from earlier:= [ (50k²h)/(16 + 25k²) ) ]² - 2*(25k²h² - 400)/(16 + 25k²) - 2h*(50k²h)/(16 + 25k²) + 2h².This looks complicated, but let's compute each term step by step.First term: [ (50k²h)/(16 + 25k²) ]² = (2500k⁴h²)/(16 + 25k²)^2.Second term: -2*(25k²h² - 400)/(16 + 25k²) = (-50k²h² + 800)/(16 + 25k²).Third term: -2h*(50k²h)/(16 + 25k²) = (-100k²h²)/(16 + 25k²).Fourth term: +2h².Putting all together:= [2500k⁴h²/(16 + 25k²)^2] + [(-50k²h² + 800)/(16 + 25k²)] + [(-100k²h²)/(16 + 25k²)] + 2h².Let me combine the second and third terms since they have the same denominator:[(-50k²h² + 800) + (-100k²h²)]/(16 + 25k²) = (-150k²h² + 800)/(16 + 25k²).Therefore, the expression becomes:2500k⁴h²/(16 + 25k²)^2 + (-150k²h² + 800)/(16 + 25k²) + 2h².Now, let's write all terms over the same denominator to combine them. The first term has denominator (16 + 25k²)^2, the second term has denominator (16 + 25k²), and the third term is 2h². Let's express 2h² as 2h²*(16 + 25k²)^2/(16 + 25k²)^2). However, this might not be the best approach. Alternatively, let's combine the first two terms and then add the third term.First term: 2500k⁴h²/(16 + 25k²)^2.Second term: (-150k²h² + 800)/(16 + 25k²) = [(-150k²h² + 800)(16 + 25k²)]/(16 + 25k²)^2.Third term: 2h² = 2h²*(16 + 25k²)^2/(16 + 25k²)^2.Therefore, combining all three terms over the same denominator (16 + 25k²)^2:Numerator = 2500k⁴h² + (-150k²h² + 800)(16 + 25k²) + 2h²(16 + 25k²)^2.Let me compute each part of the numerator step by step.First part: 2500k⁴h².Second part: (-150k²h² + 800)(16 + 25k²).Multiply out:-150k²h²*16 + (-150k²h²)*25k² + 800*16 + 800*25k²= -2400k²h² - 3750k⁴h² + 12800 + 20000k².Third part: 2h²(16 + 25k²)^2.First, expand (16 + 25k²)^2:= 16² + 2*16*25k² + (25k²)^2= 256 + 800k² + 625k⁴.Then multiply by 2h²:= 2h²*256 + 2h²*800k² + 2h²*625k⁴= 512h² + 1600k²h² + 1250k⁴h².Now, combining all three parts:First part: 2500k⁴h².Second part: -2400k²h² - 3750k⁴h² + 12800 + 20000k².Third part: 512h² + 1600k²h² + 1250k⁴h².Adding all these together:For k⁴h² terms:2500k⁴h² - 3750k⁴h² + 1250k⁴h² = (2500 - 3750 + 1250)k⁴h² = 0.For k²h² terms:-2400k²h² + 1600k²h² = (-2400 + 1600)k²h² = -800k²h².For h² terms:512h².Constant terms:12800 + 20000k².So, the numerator becomes:-800k²h² + 512h² + 12800 + 20000k².Therefore, the entire expression for (x₁ - h)^2 + (x₂ - h)^2 is:[ -800k²h² + 512h² + 12800 + 20000k² ] / (16 + 25k²)^2.But wait, actually, all terms are over (16 + 25k²)^2. So, the entire numerator is as above.Therefore, the sum |PA|² + |PB|² is:(1 + k²) * [ (-800k²h² + 512h² + 12800 + 20000k² ) / (16 + 25k²)^2 ].So, this expression needs to be independent of h. That means the coefficients of h² and h must be zero. Since there is no h term (only h² and constants), we need the coefficient of h² to be zero. The constant term can remain, as it's independent of h.Therefore, let's look at the numerator inside the brackets:-800k²h² + 512h² + 12800 + 20000k².Grouping the h² terms:(-800k² + 512)h² + (12800 + 20000k²).For the entire expression |PA|² + |PB|² to be independent of h, the coefficient of h² must be zero. So:-800k² + 512 = 0.Solving for k²:-800k² + 512 = 0=> 800k² = 512=> k² = 512 / 800Simplify:Divide numerator and denominator by 16: 32 / 50.Divide numerator and denominator by 2: 16 / 25.Thus, k² = 16/25, so k = ±4/5.But we need to check if this value indeed makes the expression independent of h. Let's verify.If k² = 16/25, then k = 4/5 or -4/5. Since the problem asks for the value of k, which can be positive or negative. But let me check if both signs are acceptable.But let's verify. Let's substitute k² = 16/25 into the numerator.First, compute the coefficient of h²:-800*(16/25) + 512 = - (800*16)/25 + 512 = - (12800)/25 + 512 = -512 + 512 = 0. Perfect.Now, the constant term:12800 + 20000*(16/25) = 12800 + (20000*16)/25 = 12800 + (320000)/25 = 12800 + 12800 = 25600.Therefore, the numerator becomes 25600, so the entire expression for |PA|² + |PB|² is:(1 + k²) * [25600 / (16 + 25k²)^2 ].Since k² = 16/25, let's compute 1 + k² = 1 + 16/25 = 41/25.16 + 25k² = 16 + 25*(16/25) = 16 + 16 = 32.Therefore, the expression becomes:(41/25) * [25600 / (32)^2 ] = (41/25)*(25600 / 1024).Simplify 25600 / 1024 = 25. So, (41/25)*25 = 41.Therefore, |PA|² + |PB|² = 41, which is indeed independent of h. So, this works for both k = 4/5 and k = -4/5. However, since the problem mentions the slope k, and slope can be positive or negative. But wait, does the problem specify that k is positive? Let me check.The problem says "a line passing through P with slope k". Since slope can be positive or negative, but the problem asks for "the value of k", which might imply a specific value. However, since both 4/5 and -4/5 would satisfy the condition (because squaring would make them the same), but in the quadratic equation, the k² term would be the same. However, in the expression for the line, the sign of k affects the direction of the line.Wait, but when we derived the condition, we set the coefficient of h² to zero, which only depends on k². Therefore, both positive and negative k would satisfy the equation. However, in the problem statement, they are asking for "the value of k", which might accept both 4/5 and -4/5, but perhaps the answer is positive as the principal value. Wait, let me check if the problem has any restrictions.Looking back at the problem: "A line passing through P with slope k intersects the ellipse at points A and B. If |PA|² + |PB|² depends only on k and not on P, find the value of k."So, the problem doesn't specify whether k is positive or negative, just to find the value of k. Since both positive and negative 4/5 satisfy the equation, but the problem might expect both answers. However, looking at similar problems, often they take the positive value unless specified otherwise. But let me verify if negative k is acceptable.Suppose k = -4/5. Then, the line would have a slope of -4/5, but the algebra would still hold because when we substituted y = k(x - h), the substitution doesn't depend on the sign of k. The quadratic equation in x would still have the same coefficients because k is squared. Therefore, both k = 4/5 and k = -4/5 would lead to the same result. Therefore, the answer should be k = ±4/5.Wait, but the problem says "find the value of k". So maybe both values are acceptable. But let me check the original problem again. It says "A line passing through P with slope k intersects the ellipse at points A and B." So, if the slope is negative, the line would be going downward, but the intersection points would still exist. However, the problem states "find the value of k", so perhaps there are two values. However, in the problem statement, maybe only one of them works? Wait, but from our derivation, both k = 4/5 and k = -4/5 satisfy the condition. So, the answer is k = ±4/5.But let me check once again if there's a mistake in the calculation. When we set the coefficient of h² to zero:-800k² + 512 = 0 => k² = 512/800 = 16/25. So, yes, k = ±4/5. So, the answer is k = ±4/5.However, looking back at the problem statement, the problem might be expecting a single answer. Wait, maybe there's an error in the calculation. Let me check the coefficients again.Wait, when I combined the terms for the numerator:After expanding all parts:First part: 2500k⁴h².Second part: -2400k²h² -3750k⁴h² +12800 +20000k².Third part: 512h² +1600k²h² +1250k⁴h².Then, adding all together:k⁴h²: 2500 -3750 +1250 = 0. Correct.k²h²: -2400 +1600 = -800. Correct.h²: 512.Constants: 12800 +20000k².So, the numerator is (-800k² + 512)h² + (12800 +20000k²). So, coefficient of h² is -800k² +512. Setting to zero gives k² =512/800=16/25. Correct.So, k=±4/5. Therefore, the answer should be k=±4/5.But in the problem statement, it's possible that the answer is positive 4/5. Let me check if negative slope would cause any issues. For example, maybe the line with negative slope doesn't intersect the ellipse at two real points for all positions of P? Wait, but since P is on the major axis (the x-axis from -5 to 5), and the line passes through P with slope k, as long as the line is not vertical, it will intersect the ellipse. For k = -4/5, the line is descending, but since the ellipse extends from x=-5 to x=5, and y from -4 to 4, the line should still intersect the ellipse at two points. Therefore, both slopes are acceptable.However, the problem is from an Olympiad, which sometimes expects all possible answers. But maybe in the original problem, the answer is 4/5. Let me check with an example.Suppose h=0 (P at the center). Then the line with slope k through (0,0) is y=kx. The intersection points with the ellipse x²/25 + y²/16=1. Substitute y=kx:x²/25 +k²x²/16=1 => x²(1/25 +k²/16)=1 => x²=1/(1/25 +k²/16). The points A and B would be (x, kx) and (-x, -kx). Then, PA^2 + PB^2 would be (x^2 + (kx)^2) + ((-x)^2 + (-kx)^2) = 2x²(1 +k²). Substituting x²:2*(1/(1/25 +k²/16))*(1 +k²). If k=4/5, then 1/25 + (16/25)/16 =1/25 +1/25=2/25. So, denominator is 2/25, so x²=25/2. Then PA^2 + PB^2=2*(25/2)*(1 +16/25)=25*(41/25)=41. Similarly, if k=-4/5, same result. So, indeed, for h=0, both slopes give the same result. Similarly, for h≠0, since in our general calculation, the sum is 41 regardless of h. Therefore, both k=4/5 and k=-4/5 work.However, the problem asks for "the value of k". So, unless it's specified to provide all possible values, it's possible that both 4/5 and -4/5 are correct. But since in the problem statement, the slope is just a real number, and we derived k²=16/25, so k=±4/5. However, the problem might expect both answers. But let me check if the problem is from a specific source or if there's a convention.Wait, in the original problem statement, the major axis is the x-axis. The line with slope k passing through P on the x-axis. If k is positive, the line goes up, if negative, it goes down. But since the ellipse is symmetric about both axes, both slopes would give the same result. Therefore, both values are acceptable.But in the answer, how is it presented? If the answer is a number, then multiple answers might be boxed. However, the problem says "find the value of k". Since k can take two values, but perhaps in the problem's context, they consider both values. So, the answer is k = ±4/5.However, let me check once again.Suppose k=4/5. Then, the line is y = (4/5)(x - h). Similarly, for k=-4/5, y = (-4/5)(x - h). Since the ellipse is symmetric with respect to the x-axis, reflecting the line over the x-axis would correspond to changing the sign of k. The distances PA and PB would remain the same because distance is a positive quantity. Therefore, |PA|² + |PB|² would be the same for both slopes. Hence, both values of k are valid. Therefore, the answer is k=±4/5.But in the problem statement, it might be expected to write both values. However, the problem is presented as a single answer question with boxed{} notation. Maybe they expect both answers inside the box. But in some cases, maybe the answer is 4/5. Let me check the calculation again.Wait, when we set the coefficient of h² to zero, we had -800k² +512 =0 => k²=512/800=16/25 => k=±4/5. So, both are solutions. Therefore, the answer is k=±4/5, which in boxed form would be boxed{pm dfrac{4}{5}}.Alternatively, if the problem expects both answers separately, but given the way the question is phrased, "find the value of k", it's possible that both are required. Therefore, the answer is boxed{pm dfrac{4}{5}}.Final Answerboxed{pm dfrac{4}{5}}</think>Given the ellipse (frac{x^2}{25} + frac{y^2}{16} = 1) with a moving point (P(h, 0)) on the major axis, we need to find the slope (k) such that (|PA|^2 + |PB|^2) depends only on (k) and not on (h).1. Equation of the Line: The line through (P(h, 0)) with slope (k) is (y = k(x - h)).2. Intersection with Ellipse: Substitute (y = k(x - h)) into the ellipse equation: [ frac{x^2}{25} + frac{(k(x - h))^2}{16} = 1 ] Simplifying, we get: [ (16 + 25k^2)x^2 - 50k^2hx + 25k^2h^2 - 400 = 0 ]3. Vieta's Formulas: For roots (x_1) and (x_2): [ x_1 + x_2 = frac{50k^2h}{16 + 25k^2}, quad x_1x_2 = frac{25k^2h^2 - 400}{16 + 25k^2} ]4. Distances Calculation: The distances (|PA|^2) and (|PB|^2) are: [ |PA|^2 + |PB|^2 = (1 + k^2)[(x_1 - h)^2 + (x_2 - h)^2] ] Expanding and simplifying: [ (x_1 - h)^2 + (x_2 - h)^2 = (x_1 + x_2 - 2h)^2 - 2x_1x_2 + 2h^2 ]5. Independence of (h): Setting the coefficient of (h^2) to zero: [ -800k^2 + 512 = 0 implies k^2 = frac{16}{25} implies k = pm frac{4}{5} ]Thus, the value of (k) is (boxed{pm dfrac{4}{5}}).,<think>Okay, so I need to find the value of k such that the sum of the squares of the distances from point P to points A and B on the ellipse depends only on the slope k and not on the position of P on the major axis. Let me start by understanding the problem step by step.First, the given ellipse is x²/25 + y²/16 = 1. That means the semi-major axis is 5 (since 25 is under x²) and the semi-minor axis is 4. The major axis is along the x-axis, so the points on the major axis are of the form (h, 0), where h ranges from -5 to 5. So, point P is (h, 0) for some h in that interval.A line with slope k passing through P will have the equation y = k(x - h). This line intersects the ellipse at points A and B. I need to find the points of intersection, compute the distances PA and PB, square them, add them up, and then find k such that this sum doesn't depend on h. That is, the expression should simplify in a way that all terms involving h cancel out, leaving an expression that depends solely on k.Let me outline the steps:1. Find the points of intersection between the line y = k(x - h) and the ellipse x²/25 + y²/16 = 1.2. Express these intersection points in terms of h and k.3. Calculate PA² + PB² using these coordinates.4. Simplify the expression and set the coefficients of h² and h to zero to make it independent of h.5. Solve for k.Starting with step 1: Substitute y = k(x - h) into the ellipse equation.So, substituting into x²/25 + [k²(x - h)²]/16 = 1.Let me write that out:x²/25 + [k²(x² - 2hx + h²)]/16 = 1.Multiply both sides by 400 (the least common multiple of 25 and 16) to eliminate denominators:16x² + 25k²(x² - 2hx + h²) = 400.Expanding the terms:16x² + 25k²x² - 50k²hx + 25k²h² = 400.Combine like terms:(16 + 25k²)x² - 50k²hx + (25k²h² - 400) = 0.This is a quadratic equation in x. Let me denote this as:A x² + B x + C = 0,where A = 16 + 25k²,B = -50k²h,C = 25k²h² - 400.The solutions to this quadratic will give the x-coordinates of points A and B.Let me denote the roots as x₁ and x₂. Then, by Vieta's formulas:x₁ + x₂ = -B/A = (50k²h)/(16 + 25k²),x₁x₂ = C/A = (25k²h² - 400)/(16 + 25k²).Now, since points A and B lie on the line y = k(x - h), their coordinates are (x₁, k(x₁ - h)) and (x₂, k(x₂ - h)).Point P is (h, 0). So, the distance PA is the distance between (h, 0) and (x₁, k(x₁ - h)).Using the distance formula:PA² = (x₁ - h)² + [k(x₁ - h) - 0]² = (x₁ - h)² + k²(x₁ - h)² = (1 + k²)(x₁ - h)².Similarly, PB² = (1 + k²)(x₂ - h)².Therefore, PA² + PB² = (1 + k²)[(x₁ - h)² + (x₂ - h)²].Let me compute (x₁ - h)² + (x₂ - h)². Expanding each term:(x₁ - h)² + (x₂ - h)² = x₁² - 2hx₁ + h² + x₂² - 2hx₂ + h²= (x₁² + x₂²) - 2h(x₁ + x₂) + 2h².Now, x₁² + x₂² can be expressed in terms of (x₁ + x₂)² - 2x₁x₂.So, substituting:(x₁ - h)² + (x₂ - h)² = [(x₁ + x₂)² - 2x₁x₂] - 2h(x₁ + x₂) + 2h².Now, let me substitute the values from Vieta's formulas.From before:x₁ + x₂ = (50k²h)/(16 + 25k²),x₁x₂ = (25k²h² - 400)/(16 + 25k²).So,(x₁ + x₂)² = [50k²h/(16 + 25k²)]²,2x₁x₂ = 2*(25k²h² - 400)/(16 + 25k²).Plugging these into the expression:[(50k²h/(16 + 25k²))² - 2*(25k²h² - 400)/(16 + 25k²)] - 2h*(50k²h/(16 + 25k²)) + 2h².Let me compute each term step by step.First term: (x₁ + x₂)² = (50k²h)^2 / (16 + 25k²)^2.Second term: -2x₁x₂ = -2*(25k²h² - 400)/(16 + 25k²).Third term: -2h(x₁ + x₂) = -2h*(50k²h)/(16 + 25k²).Fourth term: +2h².So, putting all together:[ (2500k⁴h²)/(16 + 25k²)² - (50k²h² - 800)/(16 + 25k²) ) ] - (100k²h²)/(16 + 25k²) + 2h².Wait, let me check the second term:-2x₁x₂ = -2*(25k²h² - 400)/(16 + 25k²) = (-50k²h² + 800)/(16 + 25k²).So, the first two terms inside the square brackets are:[2500k⁴h²/(16 + 25k²)^2 + (-50k²h² + 800)/(16 + 25k²)].Then, subtract the third term:- [100k²h²/(16 + 25k²)] and then add 2h².Hmm, this seems a bit complicated. Let me write all terms over a common denominator where possible.Let me denote D = 16 + 25k² for simplicity.So, first term: 2500k⁴h² / D².Second term: (-50k²h² + 800)/D.Third term: -100k²h²/D.Fourth term: +2h².So, combining the second and third terms:[(-50k²h² + 800) - 100k²h²]/D = (-150k²h² + 800)/D.So, now the entire expression becomes:2500k⁴h²/D² + (-150k²h² + 800)/D + 2h².Now, let me express all terms over the denominator D².First term is already over D²: 2500k⁴h²/D².Second term: (-150k²h² + 800)/D = [(-150k²h² + 800) * D]/D².Third term: 2h² = 2h² * D²/D².So, combining all terms over D²:[2500k⁴h² + (-150k²h² + 800)(D) + 2h²D²]/D².But D = 16 + 25k², so substitute back:[2500k⁴h² + (-150k²h² + 800)(16 + 25k²) + 2h²(16 + 25k²)^2]/(16 + 25k²)^2.This is getting quite involved. Let me compute each term one by one.First term: 2500k⁴h².Second term: (-150k²h² + 800)(16 + 25k²).Let me expand this:= -150k²h²*16 -150k²h²*25k² + 800*16 + 800*25k²= -2400k²h² - 3750k⁴h² + 12800 + 20000k².Third term: 2h²(16 + 25k²)^2.First, compute (16 + 25k²)^2:= 16² + 2*16*25k² + (25k²)^2= 256 + 800k² + 625k⁴.Multiply by 2h²:= 512h² + 1600k²h² + 1250k⁴h².So, putting all three terms together:First term: 2500k⁴h²Second term: -2400k²h² - 3750k⁴h² + 12800 + 20000k²Third term: 512h² + 1600k²h² + 1250k⁴h²Now, combine like terms:For k⁴h²:2500k⁴h² -3750k⁴h² +1250k⁴h² = (2500 -3750 +1250)k⁴h² = 0.For k²h²:-2400k²h² +1600k²h² = (-2400 +1600)k²h² = -800k²h².For h²:512h².For constants:12800 +20000k².So, combining all:0*k⁴h² -800k²h² +512h² +12800 +20000k².So, the numerator simplifies to:(-800k²h² +512h²) +12800 +20000k².Factor h²:h²(-800k² +512) +12800 +20000k².So, the entire expression for (x₁ - h)^2 + (x₂ - h)^2 is:[ h²(-800k² +512) +12800 +20000k² ] / (16 +25k²)^2.Then, PA² + PB² = (1 +k²) * [ h²(-800k² +512) +12800 +20000k² ] / (16 +25k²)^2.Since we want PA² + PB² to be independent of h, the coefficient of h² in the numerator must be zero. That is:-800k² +512 = 0.Solving this:-800k² +512 = 0 => 800k² = 512 => k² = 512/800.Simplify:Divide numerator and denominator by 16: 512 ÷16=32, 800 ÷16=50. So, k²=32/50=16/25.Thus, k=±4/5.Wait, but 16/25 is (4/5)^2, so k=±4/5.However, we should check if this value of k also makes the remaining terms independent of h. Let's verify.Once we set k²=16/25, let's compute the remaining terms:12800 +20000k².First, compute k²=16/25:20000k²=20000*(16/25)=20000*(16)/25= (20000/25)*16=800*16=12800.So, 12800 +20000k²=12800 +12800=25600.Thus, the numerator becomes:0*h² +25600.Therefore, PA² + PB² = (1 +k²)*25600/(16 +25k²)^2.Since k²=16/25, compute 1 +k²=1 +16/25=41/25.Denominator: 16 +25k²=16 +25*(16/25)=16 +16=32. So, denominator squared=32²=1024.Thus, PA² + PB²= (41/25)*(25600)/1024.Simplify:25600/1024=25. So, 25*(41/25)=41.Thus, PA² + PB²=41, which is indeed a constant, independent of h. Therefore, k=±4/5.But the problem asks for the value of k. Since slope can be positive or negative, but the question might expect a positive value. However, maybe both are acceptable. Let me check the problem statement again.It says "find the value of k". But k could be ±4/5. Wait, but let me check the calculation again. When we set the coefficient of h² to zero, we found k²=16/25, so k=±4/5. So, both slopes would work? Let me verify with k=4/5 and k=-4/5.But slope is a direction; the line would be the same in terms of absolute value but direction different. However, since the ellipse is symmetric with respect to both axes, the distances PA and PB squared would be the same regardless of the sign of k. Wait, actually, no, because the line with slope k and -k are different unless the ellipse is a circle. But in this case, since it's an ellipse, changing the sign of the slope would result in different lines, but maybe due to the squaring, the sum PA² + PB² would still be the same.Wait, let me test with k=4/5 and k=-4/5. Suppose we take k=4/5. Then, the line is y=(4/5)(x - h). If we take k=-4/5, the line is y=(-4/5)(x - h). Due to the symmetry of the ellipse about the x-axis, reflecting the line over the x-axis would result in points A and B being reflected as well. The distances PA and PB would be the same because reflecting doesn't change distances. Therefore, PA² + PB² would be the same for k=4/5 and k=-4/5. Hence, both values are valid.However, the problem says "find the value of k". It might be expecting both values, but given that in the problem statement they say "slope k", perhaps both positive and negative slopes are acceptable, so the answer is ±4/5. But let me check again.Wait, but in the Vieta's formulas and the subsequent calculation, we found that k²=16/25, so k=±4/5. The problem may accept both, but in the original problem statement, they might have a unique answer. Wait, let me check the problem again."Find the value of k." It doesn't specify to find all possible values. However, since both positive and negative slopes satisfy the condition, the answer should include both. But maybe in the context of the problem, due to the way the line is constructed, only one of them works. Wait, let me think again.Wait, but when I calculated PA² + PB² for k=4/5, the result was 41, which is a constant. Similarly, for k=-4/5, the calculation would be the same because k is squared in all terms. So, actually, k² is 16/25, so both k=4/5 and k=-4/5 lead to the same result. Therefore, the answer is k=±4/5.But the problem says "the value of k". It might expect both values. However, the original problem might have a unique answer. Let me check the problem again.Wait, let me verify my calculation once more. When I set the coefficient of h² to zero:-800k² + 512 = 0,so k² = 512 / 800 = 16/25,so k = ±4/5.Therefore, both values are correct. So, the answer should be k=±4/5.But the problem states "the value of k". Maybe in the answer, both are acceptable, so boxed as boxed{pm dfrac{4}{5}}.However, I need to check if the problem is from a specific source where perhaps only the positive value is expected, but in general, both slopes would satisfy the condition.Alternatively, maybe there's an error in my calculation. Let me check again.Starting from the expression after expanding everything:The numerator was:-800k²h² +512h² +12800 +20000k².Setting the coefficient of h² to zero:-800k² +512=0,which gives k²=512/800=16/25,so k=±4/5.Then, substituting back, the constant term becomes:12800 +20000k²=12800 +20000*(16/25)=12800 +12800=25600.Therefore, PA² + PB²=(1 +k²)*(25600)/(16 +25k²)^2.Compute 1 +k²=1 +16/25=41/25,Denominator=16 +25*(16/25)=16 +16=32,denominator squared=1024,thus,PA² + PB²=(41/25)*(25600)/1024=(41/25)*(25)=41.So, it's indeed 41, which is constant. Therefore, both k=4/5 and k=-4/5 satisfy the condition, so the answer is k=±4/5.But in the problem statement, it's possible that they expect the answer in positive form, but since both are valid, I should present both.However, looking back at the problem statement: "A line passing through P with slope k intersects the ellipse at points A and B." The slope is k, so k can be positive or negative. Since both values work, the answer is ±4/5.But let me check once more.Wait, is there a possible mistake in the coefficient calculations? Let me go back.Original expansion steps:After substituting y = k(x - h) into the ellipse equation, we had:16x² +25k²(x² - 2hx + h²)=400.Expanding: 16x² +25k²x² -50k²hx +25k²h²=400.Then combining x² terms: (16 +25k²)x² -50k²hx + (25k²h² -400)=0.So quadratic equation coefficients:A =16 +25k²,B=-50k²h,C=25k²h² -400.Then Vieta's formulas:x₁ +x₂=50k²h/(16 +25k²),x₁x₂=(25k²h² -400)/(16 +25k²).Then, PA² + PB²=(1 +k²)[(x₁ -h)^2 + (x₂ -h)^2].Computed (x₁ -h)^2 + (x₂ -h)^2= [(x₁ +x₂)^2 -2x₁x₂] -2h(x₁ +x₂) +2h².Substituted the expressions for x₁ +x₂ and x₁x₂.After substituting, and expanding, the numerator became:[(-800k² +512)h² +12800 +20000k²]/(16 +25k²)^2.Set coefficient of h² to zero: -800k² +512=0 => k²=512/800=16/25.Then, the remaining terms gave 25600/(16 +25k²)^2 * (1 +k²)=41.So, yes, calculations are correct. Therefore, k=±4/5.But the problem might have intended a positive slope, but mathematically both are correct. Therefore, the answer is k=±4/5. However, in the answer, sometimes they expect both values. Let me check if the original problem perhaps mentions "the value of k" in singular, but in the context, it might have two values. The problem says "find the value of k", but if there are two values, both should be given.Alternatively, maybe there's a miscalculation. Wait, but when k=4/5, the slope is 4/5, which is a certain line, and when k=-4/5, it's another line. However, due to the symmetry over x-axis, maybe both yield the same result. Therefore, both are acceptable.Therefore, the answer should be boxed{pm dfrac{4}{5}}.But let me check once again. Suppose h=0, so point P is at the center. Then the line is y=kx. The points A and B would be symmetric with respect to the center. Then PA² + PB² would be 2*(distance from center to A)^2, but in that case, when P is the center, the sum PA² + PB² is constant for any line through the center. But in our problem, P is moving along the major axis, not necessarily the center. However, in our case, when k=±4/5, the sum becomes constant regardless of P's position. So, the answer is indeed ±4/5.I think that's correct.Final AnswerThe value of ( k ) is (boxed{pm dfrac{4}{5}}).</think>Given the ellipse (frac{x^2}{25} + frac{y^2}{16} = 1), we need to find the value of (k) such that the sum (|PA|^2 + |PB|^2) depends only on (k) and not on the position of (P) on the major axis. 1. Equation of the Line and Ellipse Intersection: - The line passing through (P(h, 0)) with slope (k) is (y = k(x - h)). - Substituting (y = k(x - h)) into the ellipse equation: [ frac{x^2}{25} + frac{k^2(x - h)^2}{16} = 1 ] - Multiplying through by 400 to clear denominators: [ 16x^2 + 25k^2(x^2 - 2hx + h^2) = 400 ] - Simplifying: [ (16 + 25k^2)x^2 - 50k^2hx + 25k^2h^2 - 400 = 0 ]2. Vieta's Formulas: - Let (x_1) and (x_2) be the roots. Then: [ x_1 + x_2 = frac{50k^2h}{16 + 25k^2}, quad x_1x_2 = frac{25k^2h^2 - 400}{16 + 25k^2} ]3. Distance Calculation: - The distances (PA) and (PB) are calculated as: [ PA^2 = (1 + k^2)(x_1 - h)^2, quad PB^2 = (1 + k^2)(x_2 - h)^2 ] - Sum of squares: [ PA^2 + PB^2 = (1 + k^2)left[(x_1 - h)^2 + (x_2 - h)^2right] ] - Expanding and simplifying: [ (x_1 - h)^2 + (x_2 - h)^2 = (x_1 + x_2 - 2h)^2 - 2x_1x_2 + 2h^2 ] - Substituting Vieta's formulas and simplifying, we find the coefficient of (h^2) must be zero for independence from (h): [ -800k^2 + 512 = 0 implies k^2 = frac{16}{25} implies k = pm frac{4}{5} ]4. Verification: - Substituting (k = pm frac{4}{5}) confirms the remaining terms are constant, independent of (h).Thus, the value of (k) is (boxed{pm dfrac{4}{5}}).]

🔑:19. Let Aleft(x_{1}, y_{1}right), Bleft(x_{2}, y_{2}right), and the equation of the line passing through point P with slope k be y=k(x-a).begin{array}{l}text { From }left{begin{array}{l}y=k(x-a), frac{x^{2}}{25}+frac{y^{2}}{16}=1,end{array}right. text { we get } x_{1}+x_{2}=frac{50 a k^{2}}{16+25 k^{2}}, x_{1} x_{2}=frac{25 a^{2} k^{2}-400}{16+25 k^{2}} . text { Therefore, }|P A|^{2}+|P B|^{2} =left(1+k^{2}right)left[left(x_{1}-aright)^{2}+left(x_{2}-aright)^{2}right] =left(k^{2}+1right) frac{left(512-800 k^{2}right) a^{2}+800left(16+25 k^{2}right)}{left(16+25 k^{2}right)^{2}} .end{array}Let 512-800 k^{2}=0. Solving for k gives k= pm frac{4}{5}.

🧠:[<think>Okay, let's see here. I need to figure out which of the given approximations for the number 0.055049 is incorrect when using the rounding method. The options are A through D, each rounding to a different place value. Hmm, let's take them one by one. First, let me recall the standard rules for rounding. If the digit after the target place is 5 or more, you round up the target digit by 1; if it's less than 5, you leave the target digit as is. Also, any digits after the target place become zeros, but if they're after the decimal point, they can be dropped if they're trailing zeros. Alright.Let's start with option A: 0.1 (rounded to 0.1). So, rounding to the nearest tenth. The original number is 0.055049. The tenths place is the first digit after the decimal, which is 0. The digit after that is 5, which is in the hundredths place. Since we're rounding to the tenths place, we look at the hundredths place digit. If it's 5 or more, we round up the tenths place. Here, it's 5, so we should round up the 0 to 1. Therefore, 0.1. That seems correct. So A is okay.Moving on to option B: 0.06 (rounded to the hundredth place). The hundredth place is the second decimal digit. The original number is 0.055049. The hundredth place is the second 5. The digit after that is 5 in the thousandth place. Wait, no. Let me check again. The number is 0.055049. Breaking it down: 0.0 (tenths), 5 (hundredths), 5 (thousandths), 0 (ten-thousandths), 4 (hundred-thousandths), 9 (millionths). So if we're rounding to the hundredth place, which is the second decimal digit (5), we look at the next digit, which is the thousandth place (5). Since that's 5, we round up the hundredth place. So 5 in the hundredth place becomes 6. Therefore, 0.06. That seems right. So B is correct.Option C: 0.055 (rounded to the thousandth place). The thousandth place is the third decimal digit. Original number: 0.055049. The thousandth digit is 5. The next digit (ten-thousandth place) is 0. Since 0 is less than 5, we don't round up the thousandth place. Therefore, it should stay 5, and the rest get dropped. So 0.055. That's correct. So C is okay.Option D: 0.055 (rounded to 0.0001). Hmm, rounding to the nearest 0.0001 is the same as rounding to the ten-thousandth place. Let's confirm. The number is 0.055049. The ten-thousandth place is the fourth decimal digit. Let's see: 0.055049. Tenths: 0, hundredths: 5, thousandths: 5, ten-thousandths: 0, hundred-thousandths: 4, millionths: 9. So, rounding to the ten-thousandth place, which is 0. The next digit after that is 4 (in the hundred-thousandth place). Since 4 is less than 5, we leave the ten-thousandth place as 0. Therefore, rounding 0.055049 to the ten-thousandth place should be 0.0550 (since we need to keep the fourth decimal place). But the option says 0.055. Wait, but 0.055 is equivalent to 0.0550 if we consider trailing zeros after the decimal. However, typically when rounding to a specific decimal place, you should keep the precision up to that place. So if rounding to the ten-thousandth place (four decimal places), the result should be 0.0550, not 0.055, which is three decimal places. Therefore, D is incorrect because it only shows three decimal places when it should have four. But wait, in the problem statement, the options are written as "0.055 (rounded to 0.0001)". Let me parse that again. Rounding to 0.0001 means rounding to the nearest ten-thousandth. So, the correct rounded number should have four decimal places. However, 0.055 is written as three decimal places. But 0.055 is actually 0.0550 when considering the fourth decimal place, since 0.055 is the same as 0.0550. But is that how it's presented here? The option D says 0.055, which is three decimal places. So maybe that's incorrect because when you round to the ten-thousandth place, you need to have four decimal digits. Let me check again. The original number is 0.055049. When rounding to the ten-thousandth place (fourth decimal digit), which is 0 here. The next digit is 4, so we keep it 0. So the rounded number is 0.0550. But 0.0550 can be written as 0.055 if you drop the trailing zero. Wait, but in mathematics, when you specify rounding to a certain decimal place, you should keep the trailing zeros to indicate the precision. So 0.0550 is rounded to the ten-thousandth place, whereas 0.055 is rounded to the thousandth place. Therefore, option D is incorrect because it drops the trailing zero and presents it as 0.055, which is only three decimal places. Therefore, D is the incorrect approximation. So the answer should be D.Final Answerboxed{D}</think>To determine which approximation of 0.055049 is incorrect, we analyze each option based on the rounding rules:- Option A: 0.1 (rounded to 0.1) Rounding to the tenths place (first decimal digit). The tenths place is 0, and the next digit is 5. Since 5 is 5 or more, we round up the tenths place to 1. This gives 0.1, which is correct.- Option B: 0.06 (rounded to the hundredth place) Rounding to the hundredths place (second decimal digit). The hundredths place is 5, and the next digit is 5. Since 5 is 5 or more, we round up the hundredths place to 6. This gives 0.06, which is correct.- Option C: 0.055 (rounded to the thousandth place) Rounding to the thousandths place (third decimal digit). The thousandths place is 5, and the next digit is 0. Since 0 is less than 5, we keep the thousandths place as 5. This gives 0.055, which is correct.- Option D: 0.055 (rounded to 0.0001) Rounding to the ten-thousandths place (fourth decimal digit). The ten-thousandths place is 0, and the next digit is 4. Since 4 is less than 5, we keep the ten-thousandths place as 0. The correct rounded value should be 0.0550 (four decimal places). However, the option states 0.055 (three decimal places), which is incorrect as it drops the trailing zero necessary to indicate precision to the ten-thousandths place.Thus, the incorrect approximation is (boxed{D}).,<think>Okay, let me try to figure out which of these approximations is incorrect. The problem is asking which option doesn't correctly round 0.055049 using the specified rounding method. The options are A through D, each with different rounding places. Let me go through each one step by step.First, I need to recall how rounding works. Rounding typically involves looking at the digit immediately after the place you're rounding to. If that digit is 5 or higher, you round up the previous digit; if it's less than 5, you leave the previous digit as is. But I should double-check each case carefully because sometimes the context can affect the rounding, especially with trailing digits.Let's start with option A: 0.1 (rounded to 0.1). So here, we're rounding to the tenths place, which is the first decimal digit. The original number is 0.055049. The tenths place is 0 (as in 0.0), and the digit after that is 5, which is the hundredths place. Wait, hold on. Rounding to the tenths place would mean we look at the hundredths place digit. The number is 0.0 (tenths) followed by 5 (hundredths). Since the hundredths digit is 5, we round up the tenths place from 0 to 1. So 0.055049 rounded to the tenths place is 0.1. That seems correct. So option A is okay.Moving on to option B: 0.06 (rounded to the hundredth place). The hundredth place is the second decimal digit. Let's look at 0.055049. The hundredth place is the second 5 (0.05). The digit after that is the thousandth place, which is 5 again. Since we're rounding to the hundredth place, we check the thousandth digit. If it's 5 or more, we round up. So, the hundredth place is 5, the next digit is 5, so we round up the hundredth place from 5 to 6. Therefore, 0.055049 rounded to the hundredth place becomes 0.06. That's correct too. So option B is fine.Option C: 0.055 (rounded to the thousandth place). The thousandth place is the third decimal digit. Let's see. The original number is 0.055049. The thousandth digit is 5 (0.055). The next digit after that is 0 (the ten-thousandth place). Wait, 0.055049: breaking it down:- 0.0 (tenths)- 5 (hundredths)- 5 (thousandths)- 0 (ten-thousandths)- 4 (hundred-thousandths)- 9 (millionths)Wait, hold on, maybe I miscounted. Let's count the decimal places carefully:1. Tenths: 02. Hundredths: 53. Thousandths: 54. Ten-thousandths: 05. Hundred-thousandths: 46. Millionths: 9So when rounding to the thousandth place (third decimal), we look at the ten-thousandth place (fourth decimal). The number is 0.055049. The third decimal is 5, and the fourth is 0. Since 0 is less than 5, we don't round up. Therefore, rounding to the thousandth place should keep the 5 as is, resulting in 0.055. So option C is correct. Hmm, but wait, the original number is 0.055049. If we are rounding to the thousandth place, which is the third decimal, we look at the fourth decimal (0). Since it's 0, which is less than 5, we leave the thousandth digit as is. So 0.055 is correct. So option C is okay.Option D: 0.055 (rounded to 0.0001). Wait, rounding to 0.0001 is the same as rounding to the ten-thousandth place. Let's confirm. The ten-thousandth place is the fourth decimal. The number is 0.055049. The fourth decimal is 0, and the next digit (hundred-thousandth place) is 4. Since we're rounding to the fourth decimal place, we look at the fifth decimal. The fifth decimal is 4, which is less than 5, so we leave the fourth decimal as is. So rounding 0.055049 to the ten-thousandth place would result in 0.0550 (since the fourth decimal is 0, and we keep it). However, the option says 0.055, which is equivalent to rounding to the thousandth place, not the ten-thousandth. Wait, 0.055 is three decimal places, but the question says "rounded to 0.0001", which is four decimal places. So if we round to the ten-thousandth place, we should have four decimal digits. The original number is 0.055049. Let me write it out:- 0.0550 (ten-thousandth place is the fourth decimal, which is 0)- The next digit is 4 (hundred-thousandth place), which is less than 5, so we don't round up the fourth decimal. Therefore, rounding to the ten-thousandth place would give 0.0550. But the option D says 0.055, which is three decimal places. That seems incorrect. Wait, but maybe there's a misunderstanding here. When they say "rounded to 0.0001", they might mean rounding to the nearest 0.0001, which is the same as four decimal places. So the correct rounding would be 0.0550, not 0.055. Therefore, option D is incorrect because it only shows three decimal places, but the correct rounding should be four. Alternatively, maybe there's a different interpretation. Let me check again.If we are rounding 0.055049 to the nearest 0.0001, which is the ten-thousandth place, then:Start with 0.055049. The ten-thousandth place is the fourth decimal: 0. The digit after that is 4. Since 4 < 5, we leave the fourth decimal as 0. So the rounded number is 0.0550 (four decimal places). However, in option D, it's written as 0.055, which is three decimal places. That's truncating after the thousandth place, not rounding to the ten-thousandth. Therefore, option D is incorrect because it doesn't include the fourth decimal place. So the correct answer should be D.But let me make sure I haven't confused the rounding levels. Let's recap:- A: Tenths (1 decimal place): 0.1 – correct- B: Hundredths (2 decimal places): 0.06 – correct- C: Thousandths (3 decimal places): 0.055 – correct- D: Ten-thousandths (4 decimal places): Should be 0.0550, but given as 0.055 – incorrectTherefore, the incorrect approximation is D.</think>To determine which approximation is incorrect, we analyze each option's rounding precision:- Option A (0.1 rounded to tenths place): The tenths place is the first decimal digit. The next digit (5 in the hundredths place) rounds the tenths digit up from 0 to 1. Result: 0.1 (correct).- Option B (0.06 rounded to hundredths place): The hundredths place (second decimal) is 5. The next digit (5 in the thousandths place) rounds the hundredths digit up from 5 to 6. Result: 0.06 (correct).- Option C (0.055 rounded to thousandths place): The thousandths place (third decimal) is 5. The next digit (0 in the ten-thousandths place) does not round up. Result: 0.055 (correct).- Option D (0.055 rounded to 0.0001): Rounding to the ten-thousandth place (fourth decimal) requires four decimal digits. The fourth digit is 0, and the subsequent digit (4) does not round it up. Correct result: 0.0550, not 0.055 (incorrect).Answer: D boxed{D}]

🔑:Solution: 0.055049 approx 0.1 (rounded to 0.1); 0.055049 approx 0.06 (rounded to the hundredth place); 0.055049 approx 0.055 (rounded to the thousandth place); 0.055049 approx 0.0550 (rounded to 0.0001).Therefore, the correct choice is boxed{D}.

🧠:[<think>Alright, so I have this problem where I need to determine for which integers n ≥ 3, the sequence 1, 2, ..., n can be transformed into the reverse sequence n, n-1, ..., 1 using a specific operation. The operation allows me to select three consecutive terms a, b, c and replace them with b, c, a. I need to figure out for which n this is possible. Hmm, let me start by understanding the operation better.First, let me take a small n and see what happens. Let's try n=3 first because that's the smallest allowed value. The original sequence is 1, 2, 3. The target is 3, 2, 1. If I apply the operation once on the three elements, replacing a, b, c with b, c, a. So applying it once on 1, 2, 3 would give me 2, 3, 1. Hmm, that's not the reverse. If I apply the operation again on the new three elements? Wait, after the first operation, the sequence is 2, 3, 1. If I apply the operation again on positions 1, 2, 3, which are 2, 3, 1, replacing them with 3, 1, 2. So the sequence becomes 3, 1, 2. Still not reversed. Then apply the operation again on positions 1, 2, 3: 3, 1, 2 becomes 1, 2, 3. Wait, we're back to the original sequence. Hmm, so it cycles every three operations? That seems like a problem. For n=3, it's impossible because we can't reach the reversed sequence. So n=3 is not possible?Wait, but let me check again. The original is 1, 2, 3. Target is 3, 2, 1. The operation replaces three consecutive terms a, b, c with b, c, a. So starting with 1, 2, 3. Applying the operation once gives 2, 3, 1. Then applying it again on the same three elements (positions 1, 2, 3) gives 3, 1, 2. Then again, 1, 2, 3. So it cycles through these three permutations. None of them is 3, 2, 1. Therefore, for n=3, it's impossible.Okay, so n=3 is out. Let's check n=4. The original sequence is 1, 2, 3, 4. The target is 4, 3, 2, 1. Let's see if we can get there. Let me try some operations. First, maybe apply the operation on positions 1, 2, 3. That would change 1, 2, 3 to 2, 3, 1, resulting in 2, 3, 1, 4. Then apply the operation on positions 2, 3, 4: 3, 1, 4 becomes 1, 4, 3. So the sequence is now 2, 1, 4, 3. Then apply the operation on positions 1, 2, 3: 2, 1, 4 becomes 1, 4, 2, so the sequence is 1, 4, 2, 3. Hmm, not sure if this is getting me closer. Maybe a different approach.Alternatively, starting from 1, 2, 3, 4. Let's try moving the 4 to the front. To do that, maybe shift the first three elements. Applying the operation on 1, 2, 3 gives 2, 3, 1, 4. Then apply the operation on 2, 3, 4 (which are 3, 1, 4) to get 1, 4, 3, so the sequence becomes 2, 1, 4, 3. Then apply the operation on 1, 2, 3 (2, 1, 4) to get 1, 4, 2, 3. Hmm, not helpful. Maybe another sequence of operations. Let's try starting with positions 2, 3, 4. Original sequence: 1, 2, 3, 4. Apply to positions 2, 3, 4: replace 2, 3, 4 with 3, 4, 2. So the sequence becomes 1, 3, 4, 2. Then apply to positions 1, 2, 3: 1, 3, 4 becomes 3, 4, 1, so sequence is 3, 4, 1, 2. Then apply to positions 3, 4, 5... Wait, there is no position 5 here. Hmm. Maybe positions 2, 3, 4: 4, 1, 2 becomes 1, 2, 4. So sequence is 3, 1, 2, 4. Hmm, this seems messy. Maybe there's a smarter way to approach this.Alternatively, perhaps I can model the operation as a permutation and see what permutations are generated by these operations. For example, each operation is a permutation of three elements, shifting them cyclically. But since we can apply this operation on any three consecutive terms, the permutations generated might form a group. To reverse the entire sequence, we need to see if the reversal permutation is in that group. But group theory might be a bit abstract here. Maybe parity considerations?Wait, each operation is a cyclic permutation of three elements. A cyclic permutation of three elements is an even permutation because it can be written as two transpositions: (a b c) = (a c)(a b). Since each operation is an even permutation, composing them will result in even permutations. So the total permutation required to reverse the sequence must be an even permutation. Let's see what the reversal permutation's parity is.The reversal of a sequence of length n is the permutation that maps i to n+1 -i. The number of inversions in the reversed sequence compared to the original can be calculated. For the original sequence 1, 2, ..., n, the reversed sequence has inversions equal to n(n-1)/2. The parity of this number is the parity of the permutation. If n(n-1)/2 is even, then the permutation is even; otherwise, it's odd.So the reversal permutation is even if n(n-1)/2 is even, i.e., if n ≡ 0 or 1 mod 4. Because n(n-1)/2 is even when either n or n-1 is divisible by 4. So for n ≡ 0 or 1 mod 4, the reversal is even, otherwise, it's odd. Since each operation is an even permutation, the composition of operations can only produce even permutations. Therefore, if the reversal is an odd permutation, it's impossible. So for n ≡ 2 or 3 mod 4, reversal is impossible. But for n ≡ 0 or 1 mod 4, maybe it's possible?But wait, this is only a necessary condition, not necessarily sufficient. So even if the permutation is even, we need to ensure that the group generated by the operations includes the reversal permutation. So maybe for n ≡ 0 or 1 mod 4, it's possible? Let me check n=4. n=4 is 0 mod 4. Let's try again to reverse the sequence.Original: 1, 2, 3, 4. Target: 4, 3, 2, 1. Let's try a different approach. Maybe perform operations to move the 4 to the front step by step. Let's see:First, operate on positions 2, 3, 4: 2, 3, 4 becomes 3, 4, 2. So sequence is 1, 3, 4, 2. Then operate on positions 1, 2, 3: 1, 3, 4 becomes 3, 4, 1. So sequence is 3, 4, 1, 2. Then operate on positions 3, 4, 5... Wait, n=4, so positions 3,4,5 don't exist. Maybe positions 2,3,4: 4,1,2 becomes 1,2,4. So sequence is 3,1,2,4. Hmm. Then operate on positions 1,2,3: 3,1,2 becomes 1,2,3. So back to 1,2,3,4. Not helpful. Maybe another path.Alternatively, start with 1,2,3,4. Operate on positions 1,2,3: becomes 2,3,1,4. Then operate on positions 2,3,4: 3,1,4 becomes 1,4,3. So sequence is 2,1,4,3. Then operate on positions 1,2,3: 2,1,4 becomes 1,4,2. So sequence is 1,4,2,3. Then operate on positions 2,3,4: 4,2,3 becomes 2,3,4. So sequence is 1,2,3,4. Back again. Ugh. Maybe this is not the way.Wait, maybe there's a better strategy. Let me think about swapping adjacent elements. If I can perform a series of operations that swap two adjacent elements, then I can perform bubble sort-like operations to reverse the entire sequence. But does the given operation allow me to swap adjacent elements?Suppose I have a sequence ..., x, y, z, ... and I perform the operation on x, y, z to get y, z, x. How can this be used to swap x and y? Let me see. If I have a sequence a, b, c, applying the operation once gives b, c, a. Applying it again gives c, a, b. Applying it a third time gives a, b, c. So cycling every three operations. So it's a cyclic shift left by one each time. If we can chain these shifts to move elements around.Alternatively, think of the operation as moving the first element of the triplet to the end. So if I apply the operation on positions i, i+1, i+2, the element at position i moves to position i+2, and the others shift left. So to move an element two positions to the right. Similarly, applying the operation on the triplet before it could move an element two positions to the left? Wait, maybe.Alternatively, perhaps use multiple operations to simulate a transposition. For example, if I can perform a series of operations that swap two elements, then the parity consideration comes into play. But each operation is a cyclic shift of three elements, which is an even permutation, as it's a 3-cycle, which is even. Therefore, composing even permutations will result in even permutations. So, as mentioned before, the reversal permutation must be even. So for n ≡ 0 or 1 mod 4, reversal is even, so possible; for n ≡ 2 or 3 mod 4, reversal is odd, so impossible.But is this sufficient? For example, even if the reversal is even, does it necessarily lie in the subgroup generated by the 3-cycles of consecutive elements?Wait, the group generated by adjacent 3-cycles. In permutation group theory, the adjacent transpositions generate the symmetric group, but adjacent 3-cycles... Hmm. For n ≥ 3, the group generated by the 3-cycles (i, i+1, i+2) might be the alternating group. Since each 3-cycle is even, and the alternating group is generated by the 3-cycles. But are these specific 3-cycles sufficient to generate the entire alternating group?In general, the alternating group is generated by 3-cycles. But when we restrict to adjacent 3-cycles, is that still true? For example, in S_4, the alternating group A_4 is generated by all 3-cycles. But can it be generated by the adjacent 3-cycles (1,2,3), (2,3,4)? Let's check.In A_4, the 3-cycles (1,2,3) and (2,3,4) generate a subgroup. Let's see if we can get other 3-cycles. For example, (1,3,4). Let's see: (1,2,3)(2,3,4) = (1,2)(3,4). Hmm, that's a double transposition. Then (2,3,4)(1,2,3) = (1,3,4). Oh, so we can get (1,3,4). Then, using these, we can generate more 3-cycles. So maybe they do generate the entire A_4.If that's the case, then for n ≥ 3, the group generated by the adjacent 3-cycles is the entire alternating group. Therefore, if the reversal permutation is even (i.e., n ≡ 0 or 1 mod 4), then it can be expressed as a product of adjacent 3-cycles, hence the operations can reverse the sequence. If it's odd, then it's impossible.Therefore, the answer is all integers n ≥ 3 where n ≡ 0 or 1 mod 4. So n ≡ 0 or 1 mod 4.But let's verify this with n=4. If n=4, which is 0 mod 4, then according to this, it's possible. But when I tried earlier, I couldn't find the sequence. Maybe my trial was flawed. Let's try again more carefully.Original sequence: 1, 2, 3, 4. Target: 4, 3, 2, 1.Let me use the idea of generating the alternating group. If I can perform a series of operations equivalent to even permutations, perhaps I can reverse the sequence.Alternatively, here's a method:First, apply the operation to positions 1,2,3: 1,2,3 → 2,3,1. Now the sequence is 2,3,1,4.Next, apply the operation to positions 2,3,4: 3,1,4 → 1,4,3. Now the sequence is 2,1,4,3.Then, apply the operation to positions 1,2,3: 2,1,4 → 1,4,2. Now the sequence is 1,4,2,3.Apply the operation to positions 2,3,4: 4,2,3 → 2,3,4. Sequence becomes 1,2,3,4. Back to start. Hmm. Not helpful.Alternatively, start again:Original: 1,2,3,4.Apply to positions 1,2,3: 2,3,1,4.Apply to positions 1,2,3 again: 3,1,2,4.Apply to positions 1,2,3 again: 1,2,3,4. Back again. Not helpful.Alternatively, apply to positions 2,3,4 first:Original: 1,2,3,4 → 1,3,4,2.Then apply to positions 1,2,3: 1,3,4 → 3,4,1. So sequence is 3,4,1,2.Apply to positions 3,4,5… Not possible. Apply to positions 2,3,4: 4,1,2 → 1,2,4. So sequence is 3,1,2,4.Apply to positions 1,2,3: 3,1,2 → 1,2,3. So sequence is 1,2,3,4. Back again.Hmm, maybe there's a different approach. Let's try to use the fact that adjacent 3-cycles can generate the alternating group. If the reversal is in the alternating group, then there exists some sequence of operations to achieve it. But how?Wait, let me recall that reversing the entire sequence is equivalent to the permutation (1 n)(2 n-1)... which is a product of floor(n/2) transpositions. If n is even, then floor(n/2) = n/2 transpositions. If n is odd, then (n-1)/2 transpositions. So the parity of the reversal permutation is floor(n/2) mod 2. Wait, but earlier I thought it was n(n-1)/2 mod 2.Wait, maybe I made a mistake here. Let me recalculate the parity of the reversal permutation.The reversal permutation reverses the order of the elements. The number of inversions in the reversed sequence compared to the original is the same as the number of inversions in the sequence n, n-1, ..., 1. The number of inversions is n(n-1)/2. So the parity is n(n-1)/2 mod 2.n(n-1)/2 mod 2. Let's compute this for n=4: 4*3/2 = 6, which is 0 mod 2. So even. For n=5: 5*4/2 = 10, which is 0 mod 2. Wait, but 5 is 1 mod 4. Wait, n=5: 10 mod 2 is 0. Hmm, n=6: 6*5/2=15, which is 1 mod 2. So odd. So the parity alternates every two numbers. Wait, n=4: even, n=5: even, n=6: odd, n=7: odd, n=8: even, etc. So the parity is even when n ≡ 0 or 1 mod 4, and odd when n ≡ 2 or 3 mod 4. Which aligns with what I thought before. So reversal permutation is even if and only if n ≡ 0 or 1 mod 4.Therefore, if the group generated by the operations is the alternating group, then reversal is possible exactly when n ≡ 0 or 1 mod 4. But is the group generated by the operations the entire alternating group?From group theory, in the symmetric group S_n, the adjacent 3-cycles generate the alternating group A_n. This is a known result. For example, in the book "Permutation Groups" by Dixon and Mortimer, it's mentioned that the alternating group is generated by the 3-cycles, and since adjacent 3-cycles can be used to generate all 3-cycles through conjugation, the group generated by adjacent 3-cycles is indeed A_n. Therefore, if the reversal permutation is in A_n, i.e., even, then it can be achieved. Otherwise, it cannot.Thus, the answer is all integers n ≥ 3 where n ≡ 0 or 1 mod 4.But let me check n=4 again, maybe my earlier attempts were just not finding the right sequence.Suppose n=4. Let me look up or think of a specific sequence of operations.Start with 1, 2, 3, 4.First, apply operation on positions 1,2,3: becomes 2,3,1,4.Then apply operation on positions 2,3,4: 3,1,4 becomes 1,4,3. So now 2,1,4,3.Then apply operation on positions 1,2,3: 2,1,4 becomes 1,4,2. So 1,4,2,3.Then apply operation on positions 2,3,4: 4,2,3 becomes 2,3,4. So 1,2,3,4. Back again.Alternatively, different sequence:Original: 1,2,3,4.Apply operation on 2,3,4: becomes 1,3,4,2.Apply operation on 1,2,3: 1,3,4 becomes 3,4,1. So 3,4,1,2.Apply operation on 3,4,5... nope. Apply on 1,2,3: 3,4,1 becomes 4,1,3. So 4,1,3,2.Apply operation on 1,2,3: 4,1,3 becomes 1,3,4. So 1,3,4,2.Not helpful.Wait, maybe another path.Original: 1,2,3,4.Operation on 1,2,3: 2,3,1,4.Operation on 1,2,3 again: 3,1,2,4.Operation on 1,2,3 again: 1,2,3,4. Back.Alternatively, let's try to get 4 to the front.Original: 1,2,3,4.Apply operation on positions 2,3,4: 1,3,4,2.Apply operation on positions 1,2,3: 1,3,4 → 3,4,1. So 3,4,1,2.Apply operation on positions 1,2,3: 3,4,1 → 4,1,3. So 4,1,3,2.Apply operation on positions 3,4,5... nope. Apply on positions 2,3,4: 1,3,2 → 3,2,1. So 4,3,2,1. Wait, is that possible?Wait, let's go step by step:Starting from 4,1,3,2.Now, apply the operation on positions 2,3,4: 1,3,2 → 3,2,1. So the sequence becomes 4,3,2,1. Wait, that's the target! So yes, for n=4, it's possible.So the steps were:1. Start with 1,2,3,4.2. Apply operation on 2,3,4: 1,3,4,2.3. Apply operation on 1,2,3: 3,4,1,2.4. Apply operation on 1,2,3: 4,1,3,2.5. Apply operation on 2,3,4: 4,3,2,1.So after four operations, we achieved the reversed sequence. Therefore, n=4 is possible. So the parity argument holds here.Similarly, let's check n=5, which is 1 mod 4. Let's see if it's possible. But n=5 is a bit longer, so maybe we can rely on the group theory result. Since the reversal permutation is even (n=5: 5*4/2=10 inversions, which is even), and the operations generate the alternating group, then it's possible. So the answer should be n ≡ 0 or 1 mod 4.But to confirm, maybe test n=5 in a simpler way. Let's see:Original: 1,2,3,4,5. Target: 5,4,3,2,1.If we can perform similar steps as in n=4, moving the last element to the front step by step. For example:Start with 1,2,3,4,5.Apply operation on positions 3,4,5: 3,4,5 becomes 4,5,3. So sequence is 1,2,4,5,3.Apply operation on positions 2,3,4: 2,4,5 becomes 4,5,2. So sequence is 1,4,5,2,3.Apply operation on positions 1,2,3: 1,4,5 becomes 4,5,1. So sequence is 4,5,1,2,3.Apply operation on positions 3,4,5: 1,2,3 becomes 2,3,1. So sequence is 4,5,2,3,1.Apply operation on positions 2,3,4: 5,2,3 becomes 2,3,5. So sequence is 4,2,3,5,1.This is getting messy. Maybe another approach.Alternatively, use the method for n=4 and extend it. Since for n=4, we can reverse the sequence, maybe for n=5, we can reverse the first four elements and then adjust. But I need to be careful with overlapping operations.Alternatively, since the operations can be applied anywhere, we can reverse the sequence in segments. For example, reverse the first three elements, then shift and reverse the next three, etc. But this might not directly lead to the full reversal.Alternatively, think about the permutation required. Since the reversal is an even permutation for n=5, it can be composed of an even number of transpositions, which in turn can be composed of 3-cycles. Therefore, theoretically, it should be possible, but constructing the exact sequence of operations would be tedious.Given that the group theory argument is solid and n=4 works, I can feel more confident that the answer is n ≡ 0 or 1 mod 4. Let me check n=5 in another way.Suppose we use similar steps to n=4. To move 5 to the front:Original: 1,2,3,4,5.First, apply operation on positions 3,4,5: 3,4,5 → 4,5,3. Sequence: 1,2,4,5,3.Apply operation on positions 2,3,4: 2,4,5 → 4,5,2. Sequence: 1,4,5,2,3.Apply operation on positions 1,2,3: 1,4,5 → 4,5,1. Sequence: 4,5,1,2,3.Apply operation on positions 4,5,6... nope. Apply on positions 3,4,5: 1,2,3 → 2,3,1. Sequence: 4,5,2,3,1.Apply operation on positions 2,3,4: 5,2,3 → 2,3,5. Sequence: 4,2,3,5,1.Apply operation on positions 1,2,3: 4,2,3 → 2,3,4. Sequence: 2,3,4,5,1.Apply operation on positions 3,4,5: 4,5,1 → 5,1,4. Sequence: 2,3,5,1,4.Apply operation on positions 2,3,4: 3,5,1 → 5,1,3. Sequence: 2,5,1,3,4.This seems not helpful. Maybe a different approach.Alternatively, use the fact that adjacent 3-cycles can generate any even permutation. For example, in A_5, we can write the reversal permutation as a product of adjacent 3-cycles.But actually constructing such a product would be time-consuming. However, given that group theory tells us it's possible, and for n=4 it worked, I think we can trust the parity argument here.Therefore, the conclusion is that the sequence can be reversed if and only if n ≡ 0 or 1 mod 4. Therefore, the answer is all integers n with n ≡ 0 or 1 modulo 4.Final AnswerThe integers ( n ) are all those congruent to 0 or 1 modulo 4. Thus, the solution is (boxed{n equiv 0 text{ or } 1 pmod{4}}).</think>To determine for which integers ( n geq ǹ ) the sequence ( 1, 2, ldots, n ) can be transformed into ( n, n-1, ldots, 1 ) using the operation of replacing three consecutive terms ( a, b, c ) with ( b, c, a ), we analyze the problem as follows:1. Understanding the Operation: Each operation is a cyclic permutation of three consecutive elements, which is an even permutation. This means the group generated by these operations is a subgroup of the alternating group ( A_n ).2. Parity Consideration: The reversal of the sequence ( 1, 2, ldots, n ) corresponds to a permutation with ( frac{n(n-1)}{2} ) inversions. The parity of this permutation (even or odd) determines if it can be achieved through even permutations. The parity of ( frac{n(n-1)}{2} ) is: - Even if ( n equiv 0 ) or ( 1 pmod{4} ) - Odd if ( n equiv 2 ) or ( 3 pmod{4} )3. Group Theory: The group generated by adjacent 3-cycles (the operations given) is the alternating group ( A_n ). Therefore, the reversal permutation, if even, can be achieved through these operations.4. Verification with Examples: - For ( n = 4 ) (which is ( 0 mod 4 )), a sequence of operations successfully reverses the sequence. - For ( n = 3 ) (which is ( 3 mod 4 )), it is impossible as shown by cyclic permutations.Thus, the integers ( n ) for which the sequence can be reversed are those congruent to 0 or 1 modulo 4. The final answer is:[boxed{n equiv 0 text{ or } 1 pmod{4}}],<think>Okay, so I have this problem where I need to figure out for which integers n ≥ 3, the sequence 1, 2, ..., n can be transformed into n, n-1, ..., 1 using a specific operation. The operation allows me to select three consecutive terms a, b, c and replace them with b, c, a. I need to determine all such n where this is possible. Hmm, interesting. Let me try to break this down step by step.First, let's make sure I understand the operation correctly. If I have a sequence like [x, a, b, c, y], and I perform the operation on a, b, c, it becomes [x, b, c, a, y]. So the three consecutive elements are rotated left by one position. That is, the first element among the three moves to the end of the triplet. Got it. So each operation is a left rotation of a triplet. Now, the goal is to reverse the entire sequence using such operations. Let me start by experimenting with small values of n to see if I can notice a pattern or figure out some invariant.Starting with n=3. The original sequence is [1, 2, 3], and we want to get to [3, 2, 1]. Let's apply the operation once on the only triplet available: 1, 2, 3 becomes 2, 3, 1. Hmm, that's not the reverse. Wait, maybe another operation. But there's only one triplet again: 2, 3, 1. Applying the operation here would give 3, 1, 2. Still not the reverse. Another operation: 3, 1, 2 → 1, 2, 3. Oh no, we cycled back. So with n=3, it seems impossible? But wait, maybe I missed something. Wait, the original sequence is [1, 2, 3], target is [3, 2, 1]. Let me see:First operation on positions 1,2,3: [2, 3, 1]. Then, can I perform another operation? If I do the same operation again on positions 1,2,3: [3, 1, 2]. Then again: [1, 2, 3]. So it cycles every three operations. So with n=3, it's impossible. Wait, but the problem says "for any sequence," can we perform the operation multiple times. But maybe with overlapping triplets? Wait, but n=3, there's only one triplet. So if you can't do it with n=3, then maybe n=3 isn't possible. But let me check again. Wait, maybe I can reverse it with a single operation? But [1,2,3] becomes [2,3,1], which is not the reverse. So n=3 is impossible. So maybe even n are possible and odd n are not? Wait, let's check n=4.n=4: original sequence [1, 2, 3, 4], target [4, 3, 2, 1]. Let's try to perform some operations. Let me see.First, maybe start with the first triplet: 1,2,3. Applying the operation here gives [2,3,1,4]. Hmm. Then maybe apply the operation on the second triplet: 3,1,4. Applying the operation here gives [2,1,4,3]. Hmm. Then maybe apply the operation on the first triplet again: 2,1,4 → [1,4,2,3]. Hmm, not sure. This might take a while. Maybe there's a better way to approach this.Alternatively, perhaps look for some invariant or parity argument. Let's consider permutations. Each operation is a permutation of the sequence. Let's analyze the permutation induced by the operation.The operation replaces the triplet a, b, c with b, c, a. So in terms of permutation, if we consider the positions i, i+1, i+2, the element at position i moves to position i+2, the element at i+1 moves to i, and the element at i+2 moves to i+1. Wait, actually, let's index positions starting at 1. If we have positions 1,2,3, and apply the operation, then:Original: pos1: a, pos2: b, pos3: cAfter operation: pos1: b, pos2: c, pos3: aSo permutation: the element at position 1 goes to position 3, element at position 2 goes to position 1, element at position 3 goes to position 2.Similarly, if the triplet is at positions k, k+1, k+2, then:posk: a → posk+2posk+1: b → poskposk+2: c → posk+1So each such operation is a permutation of the three elements. Let's think about the parity of the permutation. Reversing the entire sequence is a permutation that's a reverse of the order. The parity of the reverse permutation depends on n(n-1)/2. If n ≡ 0 or 1 mod 4, then the permutation is even; otherwise, it's odd. So maybe the operations can generate only even permutations, so if the reverse is an odd permutation, then it's impossible. Let's check.Each operation is a permutation of three elements: moving a to position 3, b to 1, c to 2. That's a cyclic permutation of the three elements. A 3-cycle is an even permutation because it can be written as two transpositions. Wait, a 3-cycle is (1 3 2), which is equivalent to (1 3)(1 2). So two transpositions, hence even. Therefore, each operation is an even permutation. Therefore, the composition of even permutations is even. Therefore, the overall permutation must be even. However, reversing the entire sequence is a permutation that is a product of transpositions. How many transpositions? For reversing a sequence of length n, it's equivalent to the permutation that maps i to n+1 -i. The number of transpositions required is floor(n/2). So the parity is floor(n/2) mod 2. If floor(n/2) is even, then the permutation is even; otherwise, odd.Wait, for example, if n=4, reversing is swapping 1↔4, 2↔3. So two transpositions, which is even. For n=5, reversing is swapping 1↔5, 2↔4, and 3 stays. So two transpositions, even. Wait, n=5 would require two swaps, which is even. Wait, n=3: swap 1↔3, and 2 stays. That's one transposition, which is odd. Wait, but n=3, reversing is just a single transposition (1 3), which is odd. So for n=3, the permutation is odd. But our operations are even permutations. Therefore, if the required permutation is odd, it's impossible. So for n=3, since the reverse is an odd permutation, but we can only perform even permutations, it's impossible. For n=4, reverse is even, so it might be possible. Similarly, n=5 reverse is even (two transpositions), so possible? Wait, wait, let me check again.Wait, n=3: reverse is (1 3), which is a transposition, which is odd. Each operation is a 3-cycle, which is even. So composing even permutations can only give even permutations. Since the reverse is odd, it's impossible. For n=4: reverse is (1 4)(2 3), which is two transpositions, even. So possible. For n=5: reverse is (1 5)(2 4), two transpositions, even. So possible. For n=6: reverse is (1 6)(2 5)(3 4), three transpositions, which is odd. So n=6 reverse is odd, so impossible. So according to this, when n(n-1)/2 is even or something? Wait, maybe the parity of the reverse permutation is floor(n/2) mod 2. For n even: floor(n/2) = n/2. For example, n=4: floor(4/2)=2, even. n=6: floor(6/2)=3, odd. For n odd: floor(n/2)=(n-1)/2. For n=5: (5-1)/2=2, even. n=7: (7-1)/2=3, odd. So the parity of the reverse permutation is floor(n/2) mod 2.Therefore, if floor(n/2) is even, then the reverse permutation is even; else, odd. So when is floor(n/2) even?For n ≡ 0 mod 4: floor(n/2) = n/2, which is even if n ≡ 0 mod 4.For n ≡ 1 mod 4: floor(n/2) = (n-1)/2. If n=4k+1, then (4k+1-1)/2=2k, which is even.For n ≡ 2 mod 4: floor(n/2)=n/2 = 2k+1, which is odd.For n ≡ 3 mod 4: floor(n/2)=(n-1)/2=(4k+3 -1)/2=2k+1, which is odd.Therefore, floor(n/2) is even if and only if n ≡ 0 or 1 mod 4. Therefore, the reverse permutation is even if n ≡ 0 or 1 mod 4, and odd if n ≡ 2 or 3 mod 4. Since our operations are even permutations, the overall permutation must be even. Therefore, the reverse is achievable only if the reverse permutation is even, i.e., n ≡ 0 or 1 mod 4.Wait, but let's check with n=5. n=5 is 1 mod 4. So according to this, reverse is even, so possible. Let me try to see if it's actually possible. Maybe construct a sequence of operations.But before that, let me check n=4. Let's see if n=4 is possible. Original: 1,2,3,4. Target:4,3,2,1.Let me try to perform some operations. Let's label the positions 1,2,3,4.First operation on positions 1,2,3: 2,3,1,4.Then, operation on positions 2,3,4: 2,1,4,3. Now we have [2,1,4,3].Then, operation on positions 1,2,3: 1,4,2,3. [1,4,2,3].Then, operation on positions 2,3,4: 4,2,3,1. [1,4,2,3] → wait, if we operate on positions 2,3,4: 4,2,3 becomes 2,3,4. So the sequence becomes [1,2,3,4]. Oh, we're back to the original. Hmm, not helpful.Maybe another approach. Let's try to move the 4 to the front. How?Original: [1,2,3,4]. Let's operate on positions 2,3,4: 2,3,4 → 3,4,2. So the sequence becomes [1,3,4,2].Then operate on positions 1,2,3: 1,3,4 → 3,4,1. So [3,4,1,2].Then operate on positions 3,4,5 but wait, n=4, so positions 3,4,5 don't exist. Hmm.Alternatively, operate on positions 1,2,3 of [3,4,1,2]: 4,1,3,2.Then operate on positions 2,3,4: 1,3,2 → 3,2,1. So [4,3,2,1]. Wait, did I do that correctly?Wait, starting from [3,4,1,2], operating on positions 2,3,4: 4,1,2. Applying the operation would replace 4,1,2 with 1,2,4. So the sequence becomes [3,1,2,4].Wait, maybe I need to track more carefully. Let me write down each step.1. Start: [1,2,3,4]2. Operate on positions 2,3,4: replace 2,3,4 with 3,4,2 → [1,3,4,2]3. Operate on positions 1,2,3: replace 1,3,4 with 3,4,1 → [3,4,1,2]4. Operate on positions 1,2,3 again: replace 3,4,1 with 4,1,3 → [4,1,3,2]5. Operate on positions 2,3,4: replace 1,3,2 with 3,2,1 → [4,3,2,1]Wow, that worked! So in four operations, starting from n=4, we can reverse the sequence. So n=4 is possible. Therefore, according to our earlier parity argument, n ≡ 0 or 1 mod 4. So n=4 (0 mod 4) works. Let's check n=5 (1 mod 4). Let's try to reverse [1,2,3,4,5].This might be more complicated, but let's attempt to see if it's possible. Maybe use a similar strategy: moving the last element to the front step by step.Original: [1,2,3,4,5]First, let's try to move 5 to the front. To do that, we need to operate on triplets involving 5. Since 5 is at position 5, we can operate on positions 3,4,5 first.1. Operate on 3,4,5: 3,4,5 → 4,5,3 → [1,2,4,5,3]2. Then operate on 2,3,4: 2,4,5 → 4,5,2 → [1,4,5,2,3]3. Then operate on 1,2,3: 1,4,5 → 4,5,1 → [4,5,1,2,3]Now 5 is at position 2. Let's try to move it to position 1.4. Operate on 1,2,3: 4,5,1 → 5,1,4 → [5,1,4,2,3]5. Now 5 is at position 1. Now we need to reverse the rest [1,4,2,3]. Let's focus on the sublist [1,4,2,3].Trying to reverse this sublist. Let's apply similar operations.6. Operate on positions 2,3,4 (relative to the sublist, which is positions 2,3,4,5 in the original): 1,4,2,3 → wait, original sequence is [5,1,4,2,3]. So positions 2,3,4:1,4,2. Operating on these: 4,2,1 → [5,4,2,1,3]7. Now, the sequence is [5,4,2,1,3]. Let's operate on positions 3,4,5:2,1,3 → 1,3,2 → [5,4,1,3,2]8. Then operate on positions 4,5,6 but n=5. Wait, positions 4,5 can't form a triplet. Let's look elsewhere.9. The sequence is [5,4,1,3,2]. Let's operate on positions 1,2,3:5,4,1 → 4,1,5 → [4,1,5,3,2]10. Then operate on positions 3,4,5:5,3,2 →3,2,5 → [4,1,3,2,5]Hmm, not helpful. Maybe a different approach. Alternatively, since reversing the entire sequence is equivalent to reversing the order, perhaps we can use the operations to perform adjacent swaps or something similar. Let's see.Alternatively, notice that each operation is a cyclic shift of three elements. Maybe through a series of such shifts, we can simulate adjacent transpositions. If that's possible, then we can generate any permutation, but considering parity issues.Wait, but earlier we saw that the parity must be even. Since n=5 requires an even permutation (reverse is even), and the operations can generate even permutations, perhaps it's possible. But constructing the actual sequence is tedious. Maybe another approach is needed.Alternatively, consider that the problem is similar to sorting with certain operations. In this case, the operation is a specific 3-cycle. The question is whether the group generated by these operations is sufficient to reverse the sequence. From group theory, if the allowed operations generate the alternating group (all even permutations), then for n ≥ 3, except n=3, maybe? But for n=3, the alternating group is the same as the cyclic group of order 3, which doesn't include the transposition needed for reversing.But according to our earlier parity argument, if the reverse permutation is even, then it's achievable. For n ≡0 or 1 mod4, reverse is even. Therefore, the answer is all integers n ≥3 where n ≡0 or 1 mod4. So n=3 is 3≡3 mod4, which is excluded. n=4≡0, n=5≡1, n=6≡2, n=7≡3, n=8≡0, etc. So the answer should be all integers n where n ≡0 or 1 mod4.But let's confirm with n=5. If I can find a sequence of operations to reverse it, that would support the conclusion. Alternatively, maybe there's another invariant.Alternatively, let's consider the number of inversions. Reversing the sequence would change the number of inversions. Each operation (a,b,c)→(b,c,a) changes the number of inversions. Let's see how.Original triplet a,b,c. After operation: b,c,a.The number of inversions affected would be between a and b, a and c, and b and c.Suppose in the original triplet, a < b < c. Then after the operation, the triplet becomes b,c,a. Now, comparing with the rest of the sequence, but maybe it's complicated.Alternatively, consider that each operation is a specific permutation that might have a certain effect on the number of inversions modulo some number. But since parity is already considered, perhaps another invariant modulo something else.Alternatively, think of the problem as moving elements around. Each operation allows us to shift an element two positions to the right. For example, in the triplet a,b,c, a moves to the third position, so effectively, a can be moved two positions right each time. Similarly, c moves one position left. So perhaps elements can be "bubbled" through the sequence by moving them two positions at a time. However, moving an element two positions each time would allow even or odd permutations depending on the length.But moving an element from position i to position i+2, but when you reach the end, you might not be able to go further. For example, in n=4: to move 1 to position 4, you would need to do multiple operations:1,2,3,4 → operate on 1,2,3 → 2,3,1,4Then operate on 2,3,1 → 3,1,2,4Then operate on 3,1,2 → 1,2,3,4. Hmm, cycles again.Alternatively, use overlapping triplets to move elements further. For example, in n=5:Start with 1,2,3,4,5. Let's try to move 5 to the front.First, operate on 3,4,5: 3,4,5 →4,5,3 → [1,2,4,5,3]Then operate on 2,3,4:2,4,5 →4,5,2 → [1,4,5,2,3]Then operate on 1,2,3:1,4,5 →4,5,1 → [4,5,1,2,3]Now operate on 4,5,1 (wait, but in the sequence [4,5,1,2,3], positions 3,4,5 are 1,2,3. Operate on 3,4,5:1,2,3 →2,3,1 → [4,5,2,3,1]Now operate on 2,3,4:5,2,3 →2,3,5 → [4,2,3,5,1]Not helpful. Maybe another approach.Alternatively, notice that in order to reverse the sequence, we need to move each element from position i to position n+1 -i. For each element, the number of positions it needs to move is (n+1 -i) -i = n+1 -2i. Depending on n, this could be even or odd. If the displacement for each element is even, maybe it's possible. But displacement parity might be another invariant.For example, in n=4:Element 1 needs to move from position1 to position4: displacement +3 (odd)Element 4 needs to move from position4 to position1: displacement -3 (odd)Element2: position2 →3: displacement +1 (odd)Element3: position3→2: displacement -1 (odd)But in n=4, displacement for all elements is odd. But we were able to reverse the sequence. So displacement parity might not be an invariant.Alternatively, think of the operation as a permutation on the positions. Each operation affects three positions, cycling their contents. The key might be whether the permutation group generated by these operations acts transitively on the possible configurations, modulo some invariants.But this is getting too abstract. Let's go back to the parity argument. Since each operation is an even permutation, composing even permutations results in even permutations. The reverse permutation is even if and only if n≡0 or 1 mod4. Therefore, the answer is all integers n≥3 where n≡0 or 1 mod4.But let me check n=5 again. If the reverse permutation is even, then it should be achievable. Since n=5 is 1 mod4, it should be possible. But constructing the actual sequence is time-consuming. However, based on the parity argument, if the reverse permutation is in the alternating group (even permutations), and the operations generate a subgroup that includes the reverse permutation, then it's possible.Alternatively, if we can show that the operations generate the entire alternating group, then for n≥3, except when the target permutation is odd, it's possible. However, generating the alternating group with 3-cycles is possible. Since 3-cycles generate the alternating group. So if we can perform arbitrary 3-cycles, then we can generate any even permutation. But our operations are specific 3-cycles: adjacent triplets. Are adjacent 3-cycles sufficient to generate the entire alternating group?Yes, in fact, adjacent transpositions generate the symmetric group, and similarly, adjacent 3-cycles generate the alternating group. Therefore, for n≥3, the operations generate the alternating group, so any even permutation is achievable. Therefore, since the reverse permutation is even if and only if n≡0 or 1 mod4, those are the values of n for which it's possible.Therefore, the answer should be all integers n≥3 where n≡0 or 1 modulo4.To confirm, let's check n=5. Since it's in the allowed category, it should be possible. Let me attempt to find a sequence of operations.Start: [1,2,3,4,5]Goal: [5,4,3,2,1]Let's try to move 5 to the front first.Operate on positions 3,4,5: 3,4,5 →4,5,3 → [1,2,4,5,3]Operate on positions 2,3,4:2,4,5 →4,5,2 → [1,4,5,2,3]Operate on positions 1,2,3:1,4,5 →4,5,1 → [4,5,1,2,3]Now 5 is at position2. Let's move it to position1.Operate on positions1,2,3:4,5,1 →5,1,4 → [5,1,4,2,3]Now 5 is at front. Now need to reverse [1,4,2,3] to [3,2,4,1]. Hmm, complicated.Alternatively, focus on moving 4 to position2.Current sequence: [5,1,4,2,3]Operate on positions2,3,4:1,4,2 →4,2,1 → [5,4,2,1,3]Now operate on positions3,4,5:2,1,3 →1,3,2 → [5,4,1,3,2]Operate on positions4,5,6 (doesn't exist). Hmm. Let's try positions1,2,3:5,4,1 →4,1,5 → [4,1,5,3,2]Operate on positions3,4,5:5,3,2 →3,2,5 → [4,1,3,2,5]Operate on positions2,3,4:1,3,2 →3,2,1 → [4,3,2,1,5]Now we have [4,3,2,1,5]. Now move 5 to the end.Operate on positions3,4,5:2,1,5 →1,5,2 → [4,3,1,5,2]Operate on positions2,3,4:3,1,5 →1,5,3 → [4,1,5,3,2]Not helpful. This seems to be going in circles. Maybe another approach.Alternatively, since we can generate any even permutation, perhaps using known permutation techniques. For example, to reverse the entire sequence, we can perform a series of 3-cycles that effectively swap elements into their correct positions.Alternatively, let's consider that reversing the sequence is equivalent to composing multiple transpositions. Since we need an even number, let's pair them up.For n=5, reversing requires swapping (1,5), (2,4). So two transpositions, which is even. Let's try to perform these swaps using 3-cycles.First, swap 1 and5. To do that with 3-cycles, we can use a 3-cycle involving 1,5, and another element. For example, perform (1 5 2), then (1 2 5). But I need to see how this translates to our operations.Alternatively, use the operations to simulate a transposition. Since a transposition is odd, but we can use two 3-cycles to achieve a transposition. For example, (a b c)(a c b) = identity, but not helpful. Alternatively, (a b c)(a d e) can result in a transposition if they overlap appropriately. This is getting complex.Perhaps it's better to accept that based on the parity argument, if n≡0 or 1 mod4, then it's possible, otherwise not. Therefore, the answer is all integers n ≥3 where n ≡0 or 1 modulo4.Final AnswerThe integers ( n ) are all those congruent to 0 or 1 modulo 4. Thus, the solution is (boxed{n equiv 0 text{ or } 1 pmod{4}}).</think>To determine the integers ( n geq 3 ) such that the sequence ( 1, 2, ldots, n ) can be transformed into ( n, n-1, ldots, 1 ) using the given operation, we analyze the permutation parity induced by the operation. Each operation replaces three consecutive terms ( a, b, c ) with ( b, c, a ), which is a 3-cycle permutation. A 3-cycle is an even permutation because it can be written as two transpositions. Therefore, each operation is an even permutation. Composing even permutations results in an even permutation. The reverse permutation of the sequence ( 1, 2, ldots, n ) is even if and only if ( leftlfloor frac{n}{2} rightrfloor ) is even. Analyzing ( leftlfloor frac{n}{2} rightrfloor mod 2 ):- For ( n equiv 0 mod 4 ), ( leftlfloor frac{n}{2} rightrfloor = frac{n}{2} ) is even.- For ( n equiv 1 mod 4 ), ( leftlfloor frac{n}{2} rightrfloor = frac{n-1}{2} ) is even.- For ( n equiv 2 mod 4 ), ( leftlfloor frac{n}{2} rightrfloor = frac{n}{2} ) is odd.- For ( n equiv 3 mod 4 ), ( leftlfloor frac{n}{2} rightrfloor = frac{n-1}{2} ) is odd.Thus, the reverse permutation is even if and only if ( n equiv 0 ) or ( 1 mod 4 ). Therefore, the integers ( n ) for which the transformation is possible are those congruent to 0 or 1 modulo 4.[boxed{n equiv 0 text{ or } 1 pmod{4}}]]

🔑:3. For an integer n geqslant 3, the condition is satisfied if and only ifn equiv 0 or 1(bmod 4).On one hand, for a sequence sigma_{1}, sigma_{2}, cdots, sigma_{m} consisting of m different numbers, define the "inversion number" of the sequence:Ileft(sigma_{1}, sigma_{2}, cdots, sigma_{m}right)=midleft{(j, k) mid 1 leqslant jsigma_{k}right}.If sigma is transformed into tau by one operation as described in the problem, it is easy to verify thatI(tau)-I(sigma)=I(b c a)-I(a b c) in{-2,0,2} .In particular, the parity of I(sigma) remains unchanged after one operation. Therefore, if it is possible to transform 1,2, cdots, n into n, cdots, 2,1 through several operations, then I(1,2, cdots, n)=0 and I(n, cdots, 2,1)=frac{n(n-1)}{2} must have the same parity, i.e.,n equiv 0 or 1(bmod 4).On the other hand, prove that:When n equiv 0 or 1(bmod 4), it is possible to transform 1,2, cdots, n into n, cdots, 2,1 through several operations.Use induction on n to prove the conclusion (1).When n=4, the following operations can be used:1234 rightarrow 2314 rightarrow 2143 rightarrow 2431 rightarrow 4321.Assume n4.Notice that one operation a b c rightarrow b c a moves a from the left side of b c to the right side. First, perform this operation twice to move m-4 to the right of m-3, m-2, m-1, m, then perform this operation twice to move m-5 to the right of m-3, m-2, m-1, m, and so on, until moving 1 to the right of m-3, m-2, m-1, m, resulting in the sequence squarem-3, m-2, m-1, m, 1,2, cdots, m-4 .Using the fact that the conclusion (1) holds for n=4, the first 4 terms m-3, m-2, m-1, m can be transformed into reverse order through several operations. For the remaining m-4 terms, note thatm-4 equiv 0 or 1(bmod 4).If m-4=1, no further operations are needed; if m-4 geqslant 4, by the induction hypothesis, 1,2, cdots, m-4 can be transformed into reverse order through several operations.In summary, the conclusion (1) holds for n=m.

🧠:[<think>Okay, so I need to find the indefinite integral of sqrt(1 + x) divided by x squared times sqrt(x). Let me write that down properly to make sure I have it right. The integral is ∫ [sqrt(1 + x)] / [x² * sqrt(x)] dx. Hmm, let's see. Maybe I can simplify the denominator first. The denominator is x squared times sqrt(x). Since sqrt(x) is x^(1/2), then x² times x^(1/2) would be x^(2 + 1/2) = x^(5/2). So the integral becomes ∫ sqrt(1 + x) / x^(5/2) dx. That might be easier to handle.Now, sqrt(1 + x) is the same as (1 + x)^(1/2), so I can write the integrand as (1 + x)^(1/2) * x^(-5/2). Hmm, integrating that doesn't look straightforward. Maybe a substitution would help here. Let me think. When I see something like (1 + x) inside a square root, a common substitution is u = 1 + x. Let me try that. If u = 1 + x, then du = dx, and x = u - 1. Let me substitute that into the integral.So, replacing x with u - 1, the integrand becomes u^(1/2) * (u - 1)^(-5/2) du. Hmm, that seems more complicated. Maybe that substitution isn't the best approach here. Let me think of another substitution. Sometimes, when dealing with integrals that have sqrt(x) and sqrt(1 + x), a substitution like t = sqrt(1 + x)/sqrt(x) might help, but I'm not sure. Let me see.Alternatively, maybe I can manipulate the expression to make the substitution more obvious. Let me write x^(5/2) as x^2 * x^(1/2). Wait, that's how it was originally. Maybe if I write the integrand as (1 + x)^(1/2) / x^(5/2), and express (1 + x)/x as (1/x + 1). Let me see. So (1 + x)/x = 1/x + 1, but I have sqrt(1 + x) over x^(5/2). Maybe if I factor out x from the sqrt(1 + x). Let's see: sqrt(1 + x) = sqrt(x(1/x + 1)) = sqrt(x) * sqrt(1 + 1/x). That might be useful. Let me try that.So sqrt(1 + x) = sqrt(x) * sqrt(1 + 1/x). Then, the integrand becomes [sqrt(x) * sqrt(1 + 1/x)] / x^(5/2) = sqrt(1 + 1/x) / x^2. Because sqrt(x)/x^(5/2) is 1/x^2. So the integral simplifies to ∫ sqrt(1 + 1/x) / x^2 dx. Now, this looks like a substitution might be possible here. Let me set u = 1 + 1/x. Then du/dx = -1/x². Which is perfect because I have a 1/x² term in the integrand. Let's check:If u = 1 + 1/x, then du = -1/x² dx. Therefore, -du = 1/x² dx. The integral becomes ∫ sqrt(u) * (-du) = - ∫ u^(1/2) du. That's much simpler! Then integrating u^(1/2) gives (2/3) u^(3/2) + C. So substituting back, we get - (2/3) (1 + 1/x)^(3/2) + C. Let me check my steps again to make sure I didn't make a mistake.Starting with the substitution u = 1 + 1/x. Then du = derivative of 1 + 1/x with respect to x. The derivative of 1 is 0, and the derivative of 1/x is -1/x². So du = -1/x² dx. Then, solving for dx, we get dx = -x² du. Wait, but in the integral, we have 1/x² dx, so substituting that in terms of du, 1/x² dx becomes -du. So ∫ sqrt(u) * (-du) = - ∫ sqrt(u) du. Which integrates to - (2/3) u^(3/2) + C. Then substituting back u = 1 + 1/x, the result is - (2/3) (1 + 1/x)^(3/2) + C.But let me check if I can simplify 1 + 1/x. 1 + 1/x is (x + 1)/x. So (1 + 1/x)^(3/2) = [(x + 1)/x]^(3/2) = (x + 1)^(3/2) / x^(3/2). Therefore, the integral becomes - (2/3) (x + 1)^(3/2) / x^(3/2) + C. Let me see if that can be simplified further. Alternatively, maybe leave it as - (2/3) (1 + 1/x)^(3/2) + C. Either form is acceptable, but perhaps the original problem expects a certain form.Alternatively, going back to the substitution steps. Let me verify by differentiating the result. Let's compute the derivative of - (2/3) (1 + 1/x)^(3/2) with respect to x. The derivative is - (2/3) * (3/2) (1 + 1/x)^(1/2) * (-1/x²). The 2/3 and 3/2 cancel out, leaving -1 * (1 + 1/x)^(1/2) * (-1/x²) = (1 + 1/x)^(1/2) / x². But (1 + 1/x)^(1/2) is sqrt(1 + 1/x), which is the same as sqrt((x + 1)/x) = sqrt(x + 1)/sqrt(x). Therefore, the derivative is sqrt(x + 1)/sqrt(x) / x² = sqrt(x + 1)/(x^(5/2)), which is the original integrand. So that checks out.Therefore, the integral is - (2/3) (1 + 1/x)^(3/2) + C. Alternatively, written as - (2/3) sqrt((x + 1)/x)^3 + C. But maybe we can write it in terms of x and (x + 1). Let me see. If we write it as - (2/3) (x + 1)^(3/2) / x^(3/2) + C, which can be written as - (2/3) (x + 1)^(3/2) / (x^(3/2)) + C. Maybe we can factor out (x + 1)/x to the 1/2 power. Let's see: (x + 1)^(3/2) / x^(3/2) = [(x + 1)/x]^(3/2) = (1 + 1/x)^(3/2). So that's the same as before. Either form is correct. The answer can be expressed in different forms, but the simplest is probably - (2/3) (1 + 1/x)^(3/2) + C. However, sometimes the answer is expressed in terms of sqrt(x + 1) over x terms. Let me try to write it that way.Let's take - (2/3) * (x + 1)^(3/2) / x^(3/2) + C. This can be written as - (2/3) * (x + 1) * sqrt(x + 1) / (x^(3/2)) + C. Then, sqrt(x + 1) / x^(3/2) is sqrt(x + 1) / (x * sqrt(x)) = sqrt(x + 1) / (x^(3/2)). Alternatively, maybe factor out sqrt(x + 1)/x^(3/2). Hmm, not sure. Alternatively, factor (x + 1)/x: (x + 1)/x = 1 + 1/x. So maybe writing it as - (2/3) * (1 + 1/x)^(3/2) + C is the most simplified. But let me check if the problem expects a specific form.Looking back at the original integrand: sqrt(1 + x) / (x² * sqrt(x)) dx. The answer we found is - (2/3) (1 + 1/x)^(3/2) + C, which is equivalent to - (2/3) sqrt((1 + x)/x)^3 + C. Alternatively, that's - (2/3) (sqrt((1 + x)/x))^3. Which can be written as - (2/3) (sqrt(1 + x)/sqrt(x))^3. Then, that's - (2/3) (sqrt(1 + x))^3 / x^(3/2). Which is the same as - (2/3) (1 + x)^(3/2) / x^(3/2) + C. So either way is correct. The problem might accept either form, but maybe we can rationalize it further or write it in another way. Let me see if there's another substitution or method that leads to a different form.Alternatively, going back to the original substitution. Let's consider another substitution. Maybe set t = sqrt(1 + x). Then t² = 1 + x, so x = t² - 1. Then dx = 2t dt. Let's substitute into the integral. The integrand becomes t / [ (t² - 1)^2 * sqrt(t² - 1) ) ] * 2t dt. Let's see: sqrt(t² - 1) is (t² - 1)^(1/2). So the denominator is (t² - 1)^2 * (t² - 1)^(1/2) = (t² - 1)^(5/2). The numerator is t * 2t = 2t². So the integral becomes ∫ [2t² / (t² - 1)^(5/2) ] dt. Hmm, integrating 2t² / (t² - 1)^(5/2) dt. Not sure if that's easier, but maybe. Let's see.Alternatively, maybe using trigonometric substitution. Since we have t² - 1, which suggests a substitution like t = secθ. Let's try that. Let t = secθ, so dt = secθ tanθ dθ. Then t² - 1 = tan²θ. Then the integral becomes 2 sec²θ / (tan²θ)^(5/2) * secθ tanθ dθ. Simplify the denominator: (tan²θ)^(5/2) = tan^5θ. So integral is 2 sec²θ * secθ tanθ / tan^5θ dθ = 2 sec³θ tanθ / tan^5θ dθ = 2 sec³θ / tan^4θ dθ. Simplify sec³θ = 1/cos³θ and tan^4θ = sin^4θ / cos^4θ. So 2 * (1/cos³θ) / (sin^4θ / cos^4θ) dθ = 2 * (cosθ / sin^4θ) dθ. So integral becomes 2 ∫ cosθ / sin^4θ dθ. Let me set u = sinθ, then du = cosθ dθ. So integral becomes 2 ∫ du / u^4 = 2 ∫ u^(-4) du = 2 * (u^(-3)/(-3)) + C = -2/(3u³) + C = -2/(3 sin³θ) + C. Now, we need to revert back to t. Since t = secθ, then cosθ = 1/t, so sinθ = sqrt(1 - cos²θ) = sqrt(1 - 1/t²) = sqrt(t² - 1)/t. Therefore, sinθ = sqrt(t² - 1)/t. So sin³θ = (t² - 1)^(3/2)/t³. Therefore, -2/(3 sin³θ) + C = -2/(3 * (t² - 1)^(3/2)/t³) ) + C = -2 t³ / (3 (t² - 1)^(3/2)) ) + C. Simplify that. Since t = sqrt(1 + x), then t² = 1 + x, so t² - 1 = x. Therefore, (t² - 1)^(3/2) = x^(3/2). And t³ = (sqrt(1 + x))³ = (1 + x)^(3/2). Therefore, the integral becomes -2 (1 + x)^(3/2) / (3 x^(3/2)) ) + C, which is the same as before. So the result is the same as the previous substitution method. So whether I use substitution u = 1 + 1/x or t = sqrt(1 + x), I end up with the same answer. Therefore, the answer is correct.Therefore, the final answer is - (2/3) (1 + 1/x)^(3/2) + C. Alternatively, written as - (2/3) ( (x + 1)/x )^(3/2) + C. Which can also be written as - (2/3) (x + 1)^(3/2) / x^(3/2) + C. Any of these forms are acceptable. The problem might prefer one form over another, but since they are equivalent, I can choose the simplest one. Let me check if there's a way to write it in terms of sqrt(1 + x) over x terms. For example, (x + 1)^(3/2) / x^(3/2) = sqrt(x + 1)^3 / sqrt(x)^3 = (sqrt(x + 1)/sqrt(x))^3. Which is (sqrt( (x + 1)/x ))^3 = (sqrt(1 + 1/x))^3. So all forms are equivalent. Alternatively, the answer can be written as -2/(3x) * sqrt( (x + 1)/x ) * (x + 1) + C, but that's more complicated. The simplest form is probably - (2/3) ( (x + 1)/x )^(3/2) + C. To make it look nicer, maybe rationalize the exponent. For example, write it as - (2/3) * sqrt( (x + 1)^3 / x^3 ) + C. Which is the same as - (2/3) * sqrt( (x + 1)/x )^3 + C. But in terms of simplicity, I think the original substitution gives a compact form. So, after working through two substitution methods and verifying by differentiation, I'm confident that the integral is - (2/3) (1 + 1/x)^(3/2) + C. Therefore, that should be the correct answer.Final AnswerThe indefinite integral is boxed{ -dfrac{2}{3} left( sqrt{dfrac{1 + x}{x}} right)^3 + C }.</think>To find the indefinite integral:int frac{sqrt{1+x}}{x^{2} cdot sqrt{x}} , dxFirst, simplify the integrand. The denominator (x^2 cdot sqrt{x}) can be written as (x^{5/2}). Thus, the integral becomes:int frac{sqrt{1+x}}{x^{5/2}} , dxRewrite (sqrt{1+x}) as ((1+x)^{1/2}) and factor out (x) from the square root:sqrt{1+x} = sqrt{x left(1 + frac{1}{x}right)} = sqrt{x} cdot sqrt{1 + frac{1}{x}}Substituting this back into the integral, we get:int frac{sqrt{x} cdot sqrt{1 + frac{1}{x}}}{x^{5/2}} , dx = int frac{sqrt{1 + frac{1}{x}}}{x^2} , dxUse the substitution (u = 1 + frac{1}{x}), then (du = -frac{1}{x^2} , dx). This transforms the integral into:-int sqrt{u} , duIntegrate (sqrt{u}):-int u^{1/2} , du = -frac{2}{3} u^{3/2} + CSubstitute back (u = 1 + frac{1}{x}):-frac{2}{3} left(1 + frac{1}{x}right)^{3/2} + CRewrite the expression in terms of (x):-frac{2}{3} left(frac{x + 1}{x}right)^{3/2} + CThus, the indefinite integral is:boxed{ -dfrac{2}{3} left( sqrt{dfrac{1 + x}{x}} right)^3 + C },<think>Okay, let's see... I need to find the indefinite integral of sqrt(1 + x) divided by x squared times sqrt(x). Hmm, that seems a bit complicated, but maybe I can simplify the expression first before trying to integrate. First, let me rewrite the integrand to make it clearer. The denominator is x² times sqrt(x). Since sqrt(x) is x^(1/2), combining that with x² gives x^(2 + 1/2) = x^(5/2). So the integrand becomes sqrt(1 + x) divided by x^(5/2). The numerator is sqrt(1 + x), which is (1 + x)^(1/2). So the whole thing is (1 + x)^(1/2) / x^(5/2). Hmm, maybe substitution will help here. When I see a term like (1 + x) inside a square root, a common substitution is u = 1 + x or maybe something related to that. Let me try u = 1 + x. Then du/dx = 1, so du = dx. But I need to express everything in terms of u. Let's see, if u = 1 + x, then x = u - 1. So the denominator x^(5/2) becomes (u - 1)^(5/2). So the integral becomes ∫ sqrt(u) / (u - 1)^(5/2) du. Hmm, not sure if that's any better. Maybe there's another substitution.Alternatively, maybe a substitution that can simplify the ratio of the two square roots. Let's see, another approach is to write both numerator and denominator in terms of exponents. Let's express sqrt(1 + x) as (1 + x)^(1/2) and sqrt(x) as x^(1/2). So the integrand is (1 + x)^(1/2) / (x² * x^(1/2)) = (1 + x)^(1/2) / x^(5/2). So, (1 + x)^(1/2) * x^(-5/2). Hmm, combining terms, maybe we can write this as (1 + x)^(1/2) * x^(-5/2). Alternatively, factor out x from the (1 + x) term. Let's see, (1 + x) = x(1 + 1/x). So sqrt(1 + x) = sqrt(x(1 + 1/x)) = sqrt(x) * sqrt(1 + 1/x). Then the integrand becomes sqrt(x) * sqrt(1 + 1/x) / x^(5/2). Simplifying that, sqrt(x) divided by x^(5/2) is x^(1/2) / x^(5/2) = x^(1/2 - 5/2) = x^(-2) = 1/x². Then the integrand is sqrt(1 + 1/x) / x². Hmm, that seems a bit more manageable. Maybe substitution t = 1 + 1/x. Let's try that.Let t = 1 + 1/x. Then dt/dx = -1/x². So -dt = dx/x². That's promising because the integrand is sqrt(t) / x² dx. But if dx/x² is -dt, then the integral becomes ∫ sqrt(t) * (-dt) = -∫ sqrt(t) dt. Which is straightforward. Let's check that again.Wait, if t = 1 + 1/x, then dt/dx = -1/x². Therefore, dx/x² = -dt. So substituting into the integral: ∫ sqrt(t) * (dx/x²) = ∫ sqrt(t) * (-dt) = -∫ t^(1/2) dt. The integral of t^(1/2) is (2/3) t^(3/2) + C. So putting it all together, we get - (2/3) t^(3/2) + C = - (2/3) (1 + 1/x)^(3/2) + C. Then substituting back, since t = 1 + 1/x, so the integral is - (2/3) (1 + 1/x)^(3/2) + C. Let me check that again.Wait, let's verify the substitution step by step. Original integral: ∫ sqrt(1 + x) / (x² sqrt(x)) dx. After rewriting, it's ∫ sqrt(1 + x) / x^(5/2) dx. Then we factor out sqrt(x) from sqrt(1 + x), getting sqrt(x) * sqrt(1 + 1/x) / x^(5/2) = sqrt(1 + 1/x) / x². Then substitution t = 1 + 1/x, so dt/dx = -1/x². Therefore, dx/x² = -dt. Hence, the integral becomes ∫ sqrt(t) (-dt) = - ∫ t^(1/2) dt = - (2/3) t^(3/2) + C. Then substituting back t = 1 + 1/x gives - (2/3)(1 + 1/x)^(3/2) + C. But let's check if this answer is correct by differentiating it. Let me compute the derivative of - (2/3)(1 + 1/x)^(3/2). The derivative is - (2/3) * (3/2) (1 + 1/x)^(1/2) * (-1/x²). Simplifying, the 2/3 and 3/2 cancel out, leaving -1 * (1 + 1/x)^(1/2) * (-1/x²) = (1/x²) (1 + 1/x)^(1/2). Which is the same as sqrt(1 + 1/x)/x². But we need to check if this matches the original integrand. Wait, original integrand was sqrt(1 + x)/x^(5/2). Let's see if sqrt(1 + 1/x)/x² is equal to that. Let's compute sqrt(1 + x)/x^(5/2). sqrt(1 + x) is sqrt(x*(1 + 1/x)) = sqrt(x) * sqrt(1 + 1/x). So sqrt(1 + x)/x^(5/2) = sqrt(x) * sqrt(1 + 1/x) / x^(5/2) = sqrt(1 + 1/x) / x². Which is exactly the same as the derivative. So that checks out! So the integral is correct. Therefore, the solution is - (2/3) (1 + 1/x)^(3/2) + C. Alternatively, we can write this in terms of x by simplifying 1 + 1/x = (x + 1)/x. So (1 + 1/x)^(3/2) = ((x + 1)/x)^(3/2) = (x + 1)^(3/2)/x^(3/2). Therefore, the integral is - (2/3) (x + 1)^(3/2)/x^(3/2) + C. Which can be written as - (2/3) (sqrt(x + 1)/sqrt(x))^3 + C. Alternatively, factoring out sqrt(x + 1)/x^(3/2), but maybe the original substitution form is better. Alternatively, we can write it as - (2/3) ( (x + 1)/x )^(3/2 ) + C. Which is also acceptable. To make it look neater, maybe factor out (x + 1) terms. But perhaps the answer is best left as - (2/3) (1 + 1/x)^(3/2) + C. But depending on the problem's preference, maybe we can rationalize it more. Let's see:Alternatively, let's express (1 + 1/x) as (x + 1)/x. Then ( (x + 1)/x )^(3/2) is (x + 1)^(3/2) / x^(3/2). Therefore, the integral can be written as -2/3 * (x + 1)^(3/2) / x^(3/2) + C. Which might be preferable. So combining the exponents, x^(3/2) is x*sqrt(x), so (x + 1)^(3/2)/(x*sqrt(x)) = (x + 1)^(3/2)/(x^(3/2)). So we can write it as -2/(3) * ( (x + 1)/x )^(3/2) + C. Either form is acceptable, but perhaps the problem expects one form over the other. Alternatively, we can write the answer in terms of sqrt(x + 1)/x. Let's see: ( (x + 1)/x )^(3/2) = ( sqrt( (x + 1)/x ) )^3 = ( sqrt(x + 1)/sqrt(x) )^3 = (sqrt(x + 1)/sqrt(x))^3. So expanding that, it's (sqrt(x + 1))^3 / (sqrt(x))^3 = (x + 1)^(3/2)/x^(3/2). So maybe writing it as -2/(3) * (x + 1)^(3/2)/x^(3/2) + C. Alternatively, factor out (x + 1)^(1/2) from (x + 1)^(3/2), which gives (x + 1)^(1/2)*(x + 1). So then, (x + 1)^(3/2)/x^(3/2) = sqrt(x + 1)*(x + 1)/x^(3/2). But I don't know if that's any simpler.Alternatively, we can write the answer as -2/(3) * sqrt(x + 1)/x^(3/2) * (x + 1) + C. Wait, that's not necessary. Hmm.Alternatively, maybe rationalizing the expression:Let me see, (x + 1)^(3/2) is (x + 1)*sqrt(x + 1), and x^(3/2) is x*sqrt(x). So the expression becomes -2/(3) * (x + 1)*sqrt(x + 1)/(x*sqrt(x)) + C. Which can be written as -2/(3) * (x + 1)/x * sqrt( (x + 1)/x ) + C. But I think that's complicating it further.Perhaps the original answer in terms of (1 + 1/x) is simpler. Let me check both forms. Let me compute the derivative of both forms to make sure they match the integrand.First form: -2/3*(1 + 1/x)^(3/2). Derivative is -2/3 * 3/2*(1 + 1/x)^(1/2)*(-1/x²) = (1 + 1/x)^(1/2)/x², which matches the integrand sqrt(1 + x)/x^(5/2) after simplification. Second form: -2/3*(x + 1)^(3/2)/x^(3/2). Derivative is -2/3 * [ (3/2)(x + 1)^(1/2)*1 * x^(3/2) - (x + 1)^(3/2)*(3/2)x^(1/2) ] / x^3. Wait, that seems complicated. Let me compute it step by step.Let me denote y = (x + 1)^(3/2)/x^(3/2). Then dy/dx = [ derivative of numerator * denominator - numerator * derivative of denominator ] / (denominator)^2.Wait, but y is (x + 1)^(3/2) * x^(-3/2). So using the product rule: dy/dx = derivative of (x + 1)^(3/2) * x^(-3/2) + (x + 1)^(3/2) * derivative of x^(-3/2). Derivative of (x + 1)^(3/2) is (3/2)(x + 1)^(1/2). Derivative of x^(-3/2) is (-3/2)x^(-5/2). So dy/dx = (3/2)(x + 1)^(1/2) * x^(-3/2) + (x + 1)^(3/2) * (-3/2)x^(-5/2). Factor out (3/2)x^(-5/2)(x + 1)^(1/2):= (3/2)x^(-5/2)(x + 1)^(1/2) [ x^1 * 1 - (x + 1) ]= (3/2)x^(-5/2)(x + 1)^(1/2) [ x - (x + 1) ]= (3/2)x^(-5/2)(x + 1)^(1/2) [ -1 ]= - (3/2)x^(-5/2)(x + 1)^(1/2)Therefore, dy/dx = - (3/2)x^(-5/2)(x + 1)^(1/2). Therefore, the derivative of -2/3*y is -2/3 * dy/dx = -2/3 * (-3/2) x^(-5/2)(x + 1)^(1/2) ) = (1)x^(-5/2)(x + 1)^(1/2). Which is the original integrand. So that checks out as well.Therefore, both forms are correct, but perhaps the problem expects the answer in terms of x rather than 1/x. But either way, both forms are equivalent. So, the final answer is either - (2/3) (1 + 1/x)^(3/2) + C or -2/(3) * (x + 1)^(3/2)/x^(3/2) + C. Let me see if these can be simplified more.Alternatively, we can write (x + 1)^(3/2)/x^(3/2) as ((x + 1)/x)^(3/2) = (1 + 1/x)^(3/2), which brings us back to the original substitution. So they are the same. Alternatively, rationalizing further, maybe write it as -2/(3) * sqrt((x + 1)^3)/x^(3/2). Which is the same as -2/(3) * sqrt((x + 1)^3)/sqrt(x^3) = -2/(3) * sqrt( (x + 1)^3 / x^3 ) = -2/(3) * sqrt( ( (x + 1)/x )^3 ) = -2/(3) * ( (x + 1)/x )^(3/2 ), which is the same as the substitution form. Therefore, the answer can be presented in multiple equivalent forms. The simplest is probably -2/(3) * ( (x + 1)/x )^(3/2 ) + C. Alternatively, using LaTeX, that would be - frac{2}{3} left( frac{x + 1}{x} right)^{3/2} + C. But perhaps the problem expects it written in terms of square roots rather than exponents.Alternatively, expanding (x + 1)/x to 1 + 1/x, as we did before. Either way, the answer is correct. Therefore, after verifying by differentiation, I'm confident that the integral is correct. The key step was the substitution t = 1 + 1/x, which simplified the integral to a straightforward power function integral. The negative sign and the chain rule worked out correctly, and differentiating the result gives back the original integrand. So I think that's the right answer.Final AnswerThe indefinite integral is boxed{-dfrac{2}{3} left( dfrac{sqrt{1+x}}{sqrt{x}} right)^3 + C}.</think>To find the indefinite integralint frac{sqrt{1+x}}{x^{2} cdot sqrt{x}} , dx,we start by simplifying the integrand. Rewriting the denominator (x^2 cdot sqrt{x}) as (x^{5/2}), the integrand becomes:frac{sqrt{1+x}}{x^{5/2}}.Next, we factor out (sqrt{x}) from (sqrt{1+x}):sqrt{1+x} = sqrt{x left(1 + frac{1}{x}right)} = sqrt{x} cdot sqrt{1 + frac{1}{x}}.Substituting this back into the integrand, we get:frac{sqrt{x} cdot sqrt{1 + frac{1}{x}}}{x^{5/2}} = frac{sqrt{1 + frac{1}{x}}}{x^2}.We use the substitution (t = 1 + frac{1}{x}), which gives (dt = -frac{1}{x^2} , dx). Therefore, (frac{dx}{x^2} = -dt). The integral then becomes:int sqrt{t} cdot (-dt) = -int t^{1/2} , dt.Integrating (t^{1/2}), we get:-frac{2}{3} t^{3/2} + C.Substituting back (t = 1 + frac{1}{x}), we obtain:-frac{2}{3} left(1 + frac{1}{x}right)^{3/2} + C.Rewriting (1 + frac{1}{x}) as (frac{x + 1}{x}), the expression becomes:-frac{2}{3} left(frac{x + 1}{x}right)^{3/2} + C.This can be further simplified to:-frac{2}{3} left(frac{sqrt{1+x}}{sqrt{x}}right)^3 + C.Thus, the indefinite integral is:boxed{-dfrac{2}{3} left( dfrac{sqrt{1+x}}{sqrt{x}} right)^3 + C}.]

🔑:## Solutionint frac{sqrt{1+x}}{x^{2} cdot sqrt{x}} d x=int x^{-frac{5}{2}}(1+x)^{frac{1}{2}} d x=Under the integral is a binomial differential x^{m}left(a+b x^{n}right)^{p}, from whichm=-frac{5}{2} ; n=1 ; p=frac{1}{2}Since frac{m+1}{n}+p=frac{-frac{5}{2}+1}{1}+frac{1}{2}=-frac{3}{2}+frac{1}{2}=-1 is an integer, we use the substitution:a x^{-n}+b=z^{s}, text { where } s text { is the denominator of the fraction } p .In our case, the substitution is:begin{aligned}& x^{-1}+1=z^{2} & x=left(z^{2}-1right)^{-1} & d x=-1 cdotleft(z^{2}-1right)^{-2} cdot 2 z d z=-2 zleft(z^{2}-1right)^{-2} d zend{aligned}We get:begin{aligned}& =intleft(left(z^{2}-1right)^{-1}right)^{-frac{5}{2}}left(1+left(z^{2}-1right)^{-1}right)^{frac{1}{2}}(-2 z)left(z^{2}-1right)^{-2} d z= & =-2 cdot intleft(z^{2}-1right)^{frac{5}{2}}left(1+frac{1}{z^{2}-1}right)^{frac{1}{2}} zleft(z^{2}-1right)^{-2} d z= & =-2 cdot intleft(z^{2}-1right)^{frac{1}{2}}left(frac{z^{2}}{z^{2}-1}right)^{frac{1}{2}} z d z=-2 cdot intleft(z^{2}-1right)^{frac{1}{2}} frac{z}{left(z^{2}-1right)^{frac{1}{2}}} cdot z d z= & =-2 cdot int z^{2} d z=-frac{2 z^{3}}{3}+C=-frac{2left(sqrt{x^{-1}+1}right)^{3}}{3}+C=end{aligned}=-frac{2left(sqrt{frac{1}{x}+1}right)^{3}}{3}+C=-frac{2}{3}left(sqrt{frac{1+x}{x}}right)^{3}+CSource — «http://pluspi.org/wiki/index.php/������������ %D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2_%D0%98%D0%BD % mathrm{D} 1 % 82 % mathrm{D} 0 % mathrm{~B} 5 % mathrm{D} 0 % mathrm{~B} 3 % mathrm{D} 1 % 80 % mathrm{D} 0 % mathrm{~B} 0 % mathrm{D} 0 % mathrm{BB} % mathrm{D} 1 % 8 mathrm{~B}+13-10 » Categories: Kuznetsov's Problem Book Integrals Problem 13 mid IntegralsUkrainian Banner Network- Last modified: 11:46, 21 July 2009.- Content is available under CC-BY-SA 3.0.Created by GeeTeatoo## Problem Kuznetsov Integrals 13-11## Material from PlusPi

🧠:[<think>Okay, so I need to solve this vector problem. Let me read it again carefully. The problem says: Given vectors a = (√3, 1), b = (0, 1), and c = (k, √3). If vector a + 2 times vector b is perpendicular to vector c, then find the value of k. The options are A: -3, B: -2, C: 1, D: -1. Alright, let's break this down step by step.First, I remember that two vectors are perpendicular if their dot product is zero. That's a key point here. So if vector (a + 2b) is perpendicular to vector c, their dot product should equal zero. Let me verify that. Yeah, perpendicular vectors have a dot product of zero. So that's the condition I need to use here.Okay, so first I need to compute vector a + 2b. Let me do that. Vector a is (√3, 1) and vector b is (0, 1). So 2b would be 2*(0, 1) = (0, 2). Then adding that to vector a: (√3 + 0, 1 + 2) = (√3, 3). Wait, that seems straightforward. So vector a + 2b is (√3, 3). Let me double-check: √3 plus 0 is √3, and 1 plus 2 is 3. Yup, that's correct.Then vector c is given as (k, √3). So the dot product of (a + 2b) and c is (√3)(k) + (3)(√3). Because the dot product of two vectors (x1, y1) and (x2, y2) is x1*x2 + y1*y2. So applying that here: first component of (a + 2b) is √3, multiplied by k, which is the first component of c. Then the second component of (a + 2b) is 3, multiplied by the second component of c, which is √3. So that gives √3*k + 3*√3.Since the vectors are perpendicular, this dot product must be zero. Therefore, √3*k + 3*√3 = 0. Let me write that equation down:√3 * k + 3 * √3 = 0Hmm, okay. Let's solve for k. First, notice that both terms have a common factor of √3. So I can factor that out:√3 * (k + 3) = 0Now, since √3 is not zero (it's a positive irrational number), the equation simplifies to:k + 3 = 0Therefore, solving for k:k = -3So that would be option A: -3. Let me make sure I didn't make any mistakes here.Wait, let me double-check the steps. First, adding vector a and 2b: a is (√3,1), 2b is (0,2). Sum is (√3 + 0, 1 + 2) = (√3, 3). Correct. Then vector c is (k, √3). Their dot product is √3 * k + 3 * √3. Set equal to zero. Factoring out √3: √3(k + 3) = 0. Since √3 ≠ 0, then k + 3 = 0 ⇒ k = -3. That seems right. So the answer should be A: -3.But let me check the options again. The options are A: -3, B: -2, C:1, D:-1. So yes, A is -3, which matches our solution. So unless there's a mistake in the calculation, that should be the correct answer. Wait, wait a second. Let me check if I added the vectors correctly. Vector a is (√3,1), vector 2b is (0,2). So adding component-wise: x-component: √3 + 0 = √3, y-component: 1 + 2 = 3. So (√3, 3) is correct. Then vector c is (k, √3). Dot product is √3*k + 3*√3. Yes. Then setting that equal to zero. So √3k + 3√3 = 0. Divide both sides by √3 (since it's non-zero), gives k + 3 = 0 ⇒ k = -3. All steps check out. Hmm, maybe I can think of another approach to confirm. For two vectors to be perpendicular, the slope of one should be the negative reciprocal of the other. Let's try that method. Let's see if that gives the same answer.First, find the slope of vector (a + 2b). Vector (a + 2b) is (√3, 3). The slope would be rise over run, which is 3 / √3. Simplify that: 3 divided by √3 is equal to √3, because 3 / √3 = √3 * √3 / √3 = √3. So the slope of vector (a + 2b) is √3.Then, the slope of vector c is (√3 - 0)/(k - 0) if we consider it as a vector from the origin to the point (k, √3). Wait, actually, the slope would be the change in y over change in x. Since the vector c is (k, √3), assuming it's in standard position, starting at the origin, the slope would be √3 / k.If vectors are perpendicular, their slopes should be negative reciprocals. So the slope of vector (a + 2b) is √3, so the slope of vector c should be -1/√3. Therefore, √3 / k = -1/√3.Solving for k: √3 / k = -1 / √3 ⇒ Cross-multiplying: (√3)(√3) = -k ⇒ 3 = -k ⇒ k = -3. Yep, same result. So that confirms it. So k is indeed -3, which is option A. Therefore, the answer should be A.I don't see any mistakes in either method. Both the dot product approach and the slope approach lead to k = -3. So I feel confident that the correct answer is A: -3.Final Answerboxed{A}</think>Given vectors (overrightarrow{a} = (sqrt{3}, 1)), (overrightarrow{b} = (0, 1)), and (overrightarrow{c} = (k, sqrt{3})), we need to find the value of (k) such that (overrightarrow{a} + 2overrightarrow{b}) is perpendicular to (overrightarrow{c}).First, we compute (overrightarrow{a} + 2overrightarrow{b}):[overrightarrow{a} + 2overrightarrow{b} = (sqrt{3}, 1) + 2(0, 1) = (sqrt{3} + 0, 1 + 2) = (sqrt{3}, 3)]Next, we use the fact that the dot product of two perpendicular vectors is zero. The dot product of (overrightarrow{a} + 2overrightarrow{b}) and (overrightarrow{c}) must be zero:[(sqrt{3}, 3) cdot (k, sqrt{3}) = sqrt{3} cdot k + 3 cdot sqrt{3} = 0]Factoring out (sqrt{3}) from the equation:[sqrt{3}(k + 3) = 0]Since (sqrt{3} neq 0), we solve for (k):[k + 3 = 0 implies k = -3]Thus, the value of (k) is (boxed{A}).,<think>Okay, let me try to work through this problem step by step. So, we have three vectors here: vector a is (√3, 1), vector b is (0, 1), and vector c is (k, √3). The question says that the vector a + 2 times vector b is perpendicular to vector c, and we need to find the value of k. The options are A: -3, B: -2, C: 1, D: -1.First, I need to recall that if two vectors are perpendicular, their dot product is zero. So, that's the key point here. The problem is telling me that the sum of vector a and twice vector b is perpendicular to vector c. Therefore, the dot product of (a + 2b) and c should be zero.Let me start by calculating the vector a + 2b. Vector a is (√3, 1) and vector b is (0, 1). Multiplying vector b by 2 gives (0*2, 1*2) = (0, 2). Now, adding this to vector a: (√3 + 0, 1 + 2) = (√3, 3). So, the vector a + 2b is (√3, 3).Now, vector c is given as (k, √3). The next step is to compute the dot product of (a + 2b) and c. The dot product of two vectors (x1, y1) and (x2, y2) is x1*x2 + y1*y2. Applying that here, we have:(√3)*k + 3*√3 = 0Because the vectors are perpendicular, their dot product is zero. So, that equation equals zero. Now, let's solve for k.So, √3*k + 3√3 = 0. Let's factor out √3 from both terms:√3*(k + 3) = 0Since √3 is not zero (it's a positive irrational number), the equation simplifies to:k + 3 = 0Therefore, k = -3. So, looking at the options, A is -3. Therefore, the answer should be A.Wait, but let me double-check my calculations to make sure I didn't make a mistake.First, vector a + 2b: (√3, 1) + 2*(0, 1) = (√3 + 0, 1 + 2) = (√3, 3). That seems right.Vector c is (k, √3). The dot product is √3*k + 3*√3. Yes, because the first components are √3 and k, and the second components are 3 and √3. So, √3*k + 3*√3.Set equal to zero: √3*k + 3√3 = 0. Factoring √3 gives √3*(k + 3) = 0. Then, since √3 ≠ 0, k + 3 = 0, so k = -3. So, yeah, that seems correct.Looking back at the options, A is -3. So, the answer is A.But wait, just to be thorough, let me check if maybe there's a mistake in the problem statement or in my interpretation. The problem says "vector a + 2 vector b is perpendicular to vector c". So, yes, that's exactly what we did. So, the computation seems correct. So, unless there's a miscalculation here, I think the answer is A. Let me just confirm once more.Compute a + 2b:a is (√3, 1), 2b is (0*2, 1*2) = (0, 2). Adding those gives (√3 + 0, 1 + 2) = (√3, 3). Correct.Vector c is (k, √3). Dot product is (√3 * k) + (3 * √3). Set equal to zero:√3k + 3√3 = 0 → √3(k + 3) = 0 → k + 3 = 0 → k = -3. Yes, that's correct.Therefore, the answer is A: -3.Final Answerboxed{A}</think>Given vectors (overrightarrow{a} = (sqrt{3}, 1)), (overrightarrow{b} = (0, 1)), and (overrightarrow{c} = (k, sqrt{3})). We need to find the value of (k) such that (overrightarrow{a} + 2overrightarrow{b}) is perpendicular to (overrightarrow{c}).First, we calculate the vector (overrightarrow{a} + 2overrightarrow{b}):[overrightarrow{a} + 2overrightarrow{b} = (sqrt{3}, 1) + 2(0, 1) = (sqrt{3} + 0, 1 + 2) = (sqrt{3}, 3)]Next, we use the fact that if two vectors are perpendicular, their dot product is zero. The dot product of (overrightarrow{a} + 2overrightarrow{b}) and (overrightarrow{c}) must be zero:[(sqrt{3}, 3) cdot (k, sqrt{3}) = sqrt{3} cdot k + 3 cdot sqrt{3} = 0]Solving for (k):[sqrt{3}k + 3sqrt{3} = 0][sqrt{3}(k + 3) = 0]Since (sqrt{3} neq 0), we have:[k + 3 = 0][k = -3]Thus, the value of (k) is (boxed{A}).]

🔑:Since overrightarrow {a}+2 overrightarrow {b}=( sqrt {3},3), and overrightarrow {c}=(k, sqrt {3}), and since ( overrightarrow {a}+2 overrightarrow {b}) is perpendicular to overrightarrow {c}, then ( overrightarrow {a}+2 overrightarrow {b})cdot overrightarrow {c}=sqrt {3}k+3sqrt {3}=0. Therefore, k=-3. Hence, the correct choice is boxed{A}. By using the coordinate representation of the dot product of vectors, we can substitute the values to solve for k. This question mainly examines the coordinate representation of the dot product of vectors, which is a basic problem.

🧠:[<think>Okay, let's see. The problem is about finding a sufficient but not necessary condition for the inequality ( f(x) = a^{x^2 + 2x} < 1 ) to hold, given that ( a > 1 ). The options are A to D, which are different intervals for x. I need to figure out which interval is a sufficient condition, meaning that if x is in that interval, then the inequality definitely holds, but it's not necessary, so there could be x values outside that interval which also satisfy the inequality.First, since ( a > 1 ), the function ( a^y ) is an exponential function that increases as y increases. Therefore, ( a^{y} < 1 ) if and only if ( y < 0 ), because any positive exponent would give a result greater than 1, and when the exponent is 0, it's exactly 1. So for ( a^{x^2 + 2x} < 1 ), we need the exponent ( x^2 + 2x ) to be less than 0.So the inequality reduces to ( x^2 + 2x < 0 ). Let's solve this quadratic inequality. The quadratic equation ( x^2 + 2x = 0 ) factors to ( x(x + 2) = 0 ), so the roots are x = 0 and x = -2. The quadratic opens upwards because the coefficient of ( x^2 ) is positive. Therefore, the inequality ( x^2 + 2x < 0 ) holds between the roots. That is, when x is between -2 and 0. So the solution set is ( -2 < x < 0 ). Therefore, the inequality ( f(x) < 1 ) is true exactly when x is in (-2, 0). But the question asks for a sufficient but not necessary condition. Wait, if the actual solution is ( -2 < x < 0 ), then any interval that is a subset of (-2, 0) would be a sufficient condition, but not necessary. For example, if one of the options was (-1, 0), then choosing that interval would mean that whenever x is in (-1, 0), the inequality holds (which is true), but there are values outside of (-1, 0) like (-2, -1] that also satisfy the inequality. However, the options given are:A: (0, 1)B: (-1, 0)C: (-2, 0)D: (-2, 1)Wait, C is exactly the interval (-2, 0), which is the exact solution set. So if we say that C is a sufficient condition, but it's also necessary? Because if x is in (-2, 0), then f(x) < 1, and if x is not in (-2, 0), then f(x) >= 1. So in that case, C is both necessary and sufficient. But the problem states "one sufficient but not necessary condition". So maybe the answer is not C?Wait, hold on. Let me check again. The problem says "one sufficient but not necessary condition". That means the condition must guarantee that f(x) < 1, but there could be other x values outside the condition where f(x) < 1 also holds. However, if the exact solution is x in (-2, 0), then any interval larger than that would include points where the inequality doesn't hold. But the options are A, B, C, D. C is exactly (-2, 0), which is the exact solution. So if you use C as the condition, it's both necessary and sufficient, but the question asks for sufficient but not necessary. Therefore, perhaps the answer is a different interval.Wait, maybe there's a mistake here. Let me re-examine.The original inequality is ( a^{x^2 + 2x} < 1 ) with ( a > 1 ). As an exponential function with base a > 1, the inequality holds when the exponent is negative. So ( x^2 + 2x < 0 ). Solving ( x^2 + 2x < 0 ) gives x between -2 and 0. Therefore, the inequality ( f(x) < 1 ) is equivalent to x being in (-2, 0). So that interval is both necessary and sufficient. However, the question is phrased as "one sufficient but not necessary condition". So perhaps the answer is C, but maybe there is confusion here. Wait, if the question is asking for a sufficient but not necessary condition, then the answer should be an interval where all x in that interval satisfy f(x) < 1, but there are other x not in the interval that also satisfy f(x) <1. However, in this case, the exact solution is x in (-2, 0). So if we take a subset of (-2, 0), like option B: (-1, 0), then if x is in (-1, 0), it's sufficient that f(x) <1, but it's not necessary because there are other x in (-2, -1] that also work. Therefore, B is a sufficient but not necessary condition.Alternatively, if we take option C: (-2, 0), that is the exact solution, so it's both necessary and sufficient. But the problem says "sufficient but not necessary", so C can't be the answer. Then B is the answer. Let me check the options again.Options:A: (0,1)B: (-1,0)C: (-2,0)D: (-2,1)So, A is (0,1). Let's check if x is in (0,1), does f(x) <1? Let's see. If x is between 0 and 1, then x^2 + 2x would be between 0 + 0 = 0 and 1 + 2*1 = 3. So exponent is between 0 and 3. Since a >1, then a^y where y is between 0 and 3 would be between 1 and a^3. Therefore, f(x) would be greater than or equal to 1 in that interval. So A is not a valid condition. So A is incorrect.B: (-1,0). If x is between -1 and 0, then x is in (-1,0), which is a subset of (-2,0). Therefore, the exponent x^2 + 2x. Let's compute x^2 + 2x at x = -1: (-1)^2 + 2*(-1) = 1 - 2 = -1. At x = 0: 0 + 0 = 0. So in (-1,0), the exponent ranges from -1 to 0. Therefore, the exponent is negative (since x is not 0 or -1), so x^2 + 2x is between -1 and 0. Therefore, a^{x^2 + 2x} is between a^{-1} and 1. Since a >1, a^{-1} is 1/a which is less than 1. Therefore, in (-1,0), f(x) is between 1/a and 1, so it's less than 1. Therefore, if x is in (-1,0), then f(x) <1. However, there are x values in (-2, -1) where x^2 +2x is also negative. For example, take x = -1.5: x^2 +2x = (2.25) + 2*(-1.5) = 2.25 - 3 = -0.75. So exponent is -0.75, so a^{-0.75} is less than 1. Therefore, x = -1.5 is in (-2,0) and satisfies f(x) <1, but is not in (-1,0). Therefore, the interval (-1,0) is a sufficient condition but not necessary, because the inequality also holds outside of it. So B is a valid answer.C: (-2,0). As established earlier, this is the exact interval where the inequality holds, so it's both necessary and sufficient. But the question asks for a sufficient but not necessary condition, so C is not the answer.D: (-2,1). Let's check this interval. If x is between -2 and 1, then we have some parts where the exponent is negative and some where it's positive. For example, at x = 0.5: x^2 + 2x = 0.25 + 1 = 1.25. Then a^{1.25} >1, so f(x) >1. Therefore, D includes values where f(x) >=1, so it's not a valid condition. Therefore, D is incorrect.Hence, the correct answer is B: (-1,0).But wait, the options list C as (-2,0). Let me double-check. The problem states "one sufficient but not necessary condition". If C is the exact interval where the inequality holds, then using C as the condition would be both sufficient and necessary. But the question specifies "sufficient but not necessary", meaning that the condition doesn't cover all possible solutions, but if the condition is met, then the inequality is met. So, in this case, B is a subset of the solution interval, so it's sufficient but not necessary. However, C is the entire solution interval, so it's necessary and sufficient. Therefore, C is not the answer, B is. So the answer is B.Wait, but the original quadratic inequality solution is (-2,0). So the correct answer should be C? But according to the problem statement, since the question asks for a "sufficient but not necessary" condition, then C is actually necessary and sufficient, so it's not the answer. The answer is B, which is a sufficient condition (because all x in B satisfy the inequality) but not necessary (since there are x outside B that also satisfy the inequality). So yes, the answer is B. Option B is sufficient but not necessary.Therefore, the answer is boxed{B}.Final Answerboxed{B}</think>Given that ( a > 1 ) and ( f(x) = a^{x^2 + 2x} ), we need to find a sufficient but not necessary condition for ( f(x) < 1 ) to hold.First, we note that ( a^{y} < 1 ) if and only if ( y < 0 ) since ( a > 1 ). Therefore, we need the exponent ( x^2 + 2x < 0 ).Solving the quadratic inequality ( x^2 + 2x < 0 ):- The roots of the equation ( x^2 + 2x = 0 ) are ( x = 0 ) and ( x = -2 ).- The quadratic expression ( x^2 + 2x ) is negative between its roots, so the solution is ( -2 < x < 0 ).The interval ( -2 < x < 0 ) (option C) is the exact solution, making it both necessary and sufficient. However, the problem asks for a sufficient but not necessary condition. We consider the intervals given:- Option A: ( 0 < x < 1 ) is incorrect because the exponent here is positive, leading to ( f(x) geq 1 ).- Option B: ( -1 < x < 0 ) is a subset of ( -2 < x < 0 ). All ( x ) in this interval satisfy ( x^2 + 2x < 0 ), hence ( f(x) < 1 ). However, there are values outside this interval (e.g., ( -2 < x leq -1 )) that also satisfy ( f(x) < 1 ), making it sufficient but not necessary.- Option C: ( -2 < x < 0 ) is the exact solution, making it both necessary and sufficient.- Option D: ( -2 < x < 1 ) is incorrect because it includes values where the exponent is positive, leading to ( f(x) geq 1 ).Thus, the sufficient but not necessary condition is (boxed{B}).,<think>Alright, let's tackle this problem step by step. So, we have the function ( f(x) = a^{x^2 + 2x} ) where ( a > 1 ), and we need to find a sufficient but not necessary condition for ( f(x) < 1 ). The options are A through D, which are different intervals for x. First, let's recall that for an exponential function ( a^y ), if the base ( a ) is greater than 1, then ( a^y < 1 ) when ( y < 0 ). That's because any number greater than 1 raised to a negative power will be between 0 and 1. So, the key here is to find when the exponent ( x^2 + 2x ) is negative. So, the inequality we need to solve is ( x^2 + 2x < 0 ). Let me write that down:( x^2 + 2x < 0 )Hmm, this is a quadratic inequality. To solve it, I should factor the quadratic expression. Let's see:( x^2 + 2x = x(x + 2) )So, the inequality becomes:( x(x + 2) < 0 )Now, to solve this inequality, I need to find the intervals where the product of x and (x + 2) is negative. Let's find the critical points where the expression is zero:( x = 0 ) or ( x + 2 = 0 Rightarrow x = -2 )These critical points divide the real number line into three intervals:1. ( (-infty, -2) )2. ( (-2, 0) )3. ( (0, infty) )Now, let's test each interval to determine the sign of the expression ( x(x + 2) ):1. Interval ( (-infty, -2) ): Let's pick x = -3. Then, ( (-3)(-3 + 2) = (-3)(-1) = 3 ), which is positive.2. Interval ( (-2, 0) ): Let's pick x = -1. Then, ( (-1)(-1 + 2) = (-1)(1) = -1 ), which is negative.3. Interval ( (0, infty) ): Let's pick x = 1. Then, ( (1)(1 + 2) = 1*3 = 3 ), which is positive.Since we're looking for where the product is negative, the solution to ( x(x + 2) < 0 ) is ( (-2, 0) ). Therefore, the original inequality ( f(x) = a^{x^2 + 2x} < 1 ) holds when ( x ) is in ( (-2, 0) ). Now, looking at the answer choices:A: ( 0 < x < 1 )B: ( -1 < x < 0 )C: ( -2 < x < 0 )D: ( -2 < x < 1 )So, the exact solution is interval C: ( -2 < x < 0 ). However, the problem states that we need a "sufficient but not necessary" condition. Wait a second. Let me think. A sufficient condition means that if x is in that interval, then f(x) < 1 must hold, but there might be other intervals where f(x) < 1 also holds. But wait, according to our solution, the only interval where f(x) < 1 is exactly ( (-2, 0) ). So, that interval is both necessary and sufficient. Hmm, but the problem says "sufficient but not necessary." That seems contradictory. Wait, maybe I need to check again.Wait, no. Wait, if the problem is asking for a sufficient but not necessary condition, then the correct answer should be an interval where if x is in that interval, then f(x) < 1, but the interval isn't the entire set of solutions. But according to our previous analysis, the entire set of solutions is ( (-2, 0) ). So, if we take a subset of that interval, like option B: ( -1 < x < 0 ), then that would be a sufficient condition because any x in (-1, 0) is also in (-2, 0), so f(x) < 1. But it's not necessary because the solution isn't limited to (-1, 0). Similarly, if we take a superset interval that includes some parts where f(x) might not be less than 1, that would not be sufficient. Wait, but the options are A, B, C, D. Wait, let me re-examine the problem. The question says: "one sufficient but not necessary condition for f(x) < 1 to hold is..." So, they need an interval such that if x is in that interval, then f(x) <1, but the interval doesn't have to cover all possible x where f(x) <1. So, for example, even though the exact solution is C: (-2, 0), if an option is a subset of C, then that would be sufficient but not necessary. So, B: (-1,0) is a subset of C, so if x is in B, then f(x) <1, which is sufficient, but since there are other x's (in C but not in B) where f(x) <1, it's not necessary. Similarly, if there was an interval that was partially overlapping with C, but included some areas outside, but if the part outside also satisfies f(x) <1, then that could also be a candidate. Wait, but we already know the exact solution is C. So, let's check the options.Option C is exactly the interval where f(x) <1. So, if we choose C, then it is both necessary and sufficient. But the problem says "sufficient but not necessary". Therefore, the answer cannot be C. Therefore, the answer must be an interval that is a subset of C. Among the options, B is (-1,0), which is a subset. Therefore, B is sufficient but not necessary. However, wait, the problem says "one sufficient but not necessary condition". So, any subset of the solution interval is a sufficient condition, but not necessary. But the options given include B and C. Let me check.Wait, wait. Wait, maybe I made a mistake here. Let me think again. The problem is stating that the condition is sufficient but not necessary. So, a sufficient condition would be something that guarantees f(x) <1, but it's not the only way for f(x) <1 to hold. So, for example, if we take a smaller interval, like (-1, 0), then whenever x is in (-1,0), f(x) <1, which is sufficient. However, the necessary and sufficient condition is (-2,0). So, the interval (-1,0) is sufficient but not necessary. Similarly, if we take ( -1.5, -0.5 ), that would also be sufficient but not necessary. However, among the options, B is (-1,0), which is a subset of the solution interval, so that is sufficient but not necessary. Option C is the exact interval, so that's necessary and sufficient. Therefore, since the question asks for sufficient but not necessary, the answer should be B. But wait, let me check again. Wait, perhaps the problem is presented differently. Wait, perhaps I need to check if the question is about the inequality f(x) <1. So, for f(x) <1, since a>1, we need the exponent to be negative, which is x in (-2,0). So, any x outside of (-2,0) would make the exponent non-negative, so f(x) >=1. Therefore, the only interval where f(x) <1 is (-2,0). Therefore, the exact interval is (-2,0), so if we take a subset of that, like (-1,0), then that would be a sufficient condition (since being in (-1,0) is enough to make f(x) <1), but it's not necessary because there are other x's (like -1.5) in (-2,0) that also make f(x) <1. So, B is a sufficient but not necessary condition, whereas C is the necessary and sufficient condition. Therefore, the answer is B. Wait, but the options given are:A: 0 < x < 1B: -1 < x < 0C: -2 < x < 0D: -2 < x <1So, according to this, if we choose B, then x in (-1,0) is sufficient but not necessary. C is the necessary and sufficient condition, so it's not a "sufficient but not necessary". So, the answer is B? But wait, the problem says "one sufficient but not necessary condition". So, there can be multiple such conditions, but the options include C as the exact interval. Wait, but maybe there's a mistake here. Let me check again.Wait, the problem says "one sufficient but not necessary condition". So, if the exact interval where the inequality holds is C, then C is the necessary and sufficient condition. Therefore, it's not "sufficient but not necessary". However, if we choose a larger interval that includes C and some other parts where the inequality still holds, but wait, in reality, outside of C, the inequality doesn't hold. So, for example, if D: (-2,1) is considered, but for x between 0 and1, the exponent x^2 + 2x would be positive (since x is positive and x^2 +2x = x(x +2). When x is between 0 and1, x is positive, and x +2 is positive, so the exponent is positive, so a^positive is greater than1. Therefore, D includes regions where f(x) >=1, so D is not a sufficient condition. Similarly, A is 0 <x <1, which is where f(x) >1, so A is invalid. B is (-1,0), which is a subset of the solution interval C, so being in B is sufficient for f(x) <1, but not necessary because the full solution is C. Therefore, B is the correct answer. Therefore, the answer is B.Wait, but let me confirm once more. If x is in (-1,0), then x is in (-2,0), so exponent is negative, so f(x) <1. So, B is sufficient. However, even if x is not in B, like x = -1.5, which is in C but not in B, f(x) is still <1. So, B is sufficient but not necessary. Therefore, the answer is B. But the system initially thought the answer was C. Wait, but according to the problem statement, the question is about a sufficient but not necessary condition. So, C is the exact condition, so it's necessary and sufficient, so C is not the answer. The answer must be B. Therefore, the correct answer is B.Wait, but wait. Wait, let me check the definition again. A sufficient condition means that if the condition is met, then the result follows. But the result can also follow without the condition being met. So, in this case, if the condition is x ∈ (-1,0), then f(x) <1, which is sufficient. However, the result can also hold when x is, say, -1.5, which is not in (-1,0). Therefore, B is sufficient but not necessary. However, the interval C is exactly where the result holds, so C is necessary and sufficient. Therefore, since the problem asks for a sufficient but not necessary condition, the answer should be B. But why is the interval C an option? Maybe there's a trick here. Wait, perhaps the question is in Chinese and the translation is off? Wait, no. The original problem is in English. Hmm.Alternatively, maybe the answer is C, but let me think again. If C is (-2,0), which is the exact interval where f(x) <1. Then, is C a sufficient condition? Yes, because if x is in C, then f(x) <1. But is it necessary? Yes, because if x is not in C, then f(x) >=1. Therefore, C is both necessary and sufficient. Therefore, C cannot be the answer because the problem asks for a sufficient but not necessary condition. So, the answer must be B. Therefore, I think the answer is B. But let me check once more.Wait, let's take an example. Let a = 2, which is greater than 1. Then, f(x) = 2^{x² + 2x}. Let's check for x in (-2,0). Take x = -1: exponent is (-1)^2 + 2*(-1) = 1 -2 = -1, so f(-1) = 2^{-1} = 0.5 <1. Take x = -0.5: exponent is 0.25 + 2*(-0.5) = 0.25 -1 = -0.75, so f(-0.5)=2^{-0.75}≈0.59 <1. Take x = -1.5: exponent is (2.25) + 2*(-1.5)=2.25 -3 = -0.75, f(-1.5)=2^{-0.75}≈0.59 <1. So, all of C works. If we take x in B: (-1,0). Let's take x = -0.5: works. If we take x = -0.1: exponent is 0.01 + (-0.2)= -0.19, so f(x)=2^{-0.19}≈0.87 <1. So, B works. However, x = -1.5 is in C but not in B, and also works. So, B is a subset of C, so sufficient but not necessary. Therefore, answer is B. Therefore, the answer is B.But wait, but in the options, C is also present. But according to the definitions, a sufficient condition is one that ensures the result, but the result can occur without the condition. Therefore, C is necessary and sufficient, so not "sufficient but not necessary". Therefore, the answer must be B. So, why does the initial thought process consider C? Because maybe there's confusion between necessary and sufficient. But given the problem statement, the answer is B. However, looking back at the original problem statement, it says "one sufficient but not necessary condition". So, C is necessary and sufficient, so it's not the answer. The answer is B.But wait, maybe I need to check the problem statement again. The problem says "one sufficient but not necessary condition for f(x) < 1 to hold is". So, the condition is on x. So, the interval is the condition. Therefore, a sufficient condition is an interval such that if x is in that interval, then f(x) <1. But the condition is not necessary, meaning that there are x's not in that interval where f(x) <1 still holds. So, the only such interval among the options is B: (-1,0). Because if x is in (-1,0), then f(x) <1, but f(x) <1 also holds for x in (-2, -1]. Therefore, B is sufficient but not necessary. Therefore, the answer is B.But wait, let me check the other options. Option D: (-2,1). If x is in (-2,1), does that ensure f(x) <1? Well, for x between 0 and1, the exponent x² +2x is positive, so a^{positive} >1, since a>1. Therefore, in D, when x is between 0 and1, f(x) >1. So, D is not a sufficient condition. Similarly, A: 0<x<1 would make f(x) >1, so A is invalid. So, only B and C are candidates. C is the exact interval, so it's necessary and sufficient. Therefore, B is the answer.Therefore, after this long reasoning, I think the correct answer is B.Final Answerboxed{B}</think>Given that ( a > 1 ) and ( f(x) = a^{x^2 + 2x} ), we need to find a sufficient but not necessary condition for ( f(x) < 1 ). To solve this, we note that ( a^y < 1 ) when ( y < 0 ) since ( a > 1 ). Therefore, we need to find when the exponent ( x^2 + 2x < 0 ).Solving the quadratic inequality ( x^2 + 2x < 0 ):[x(x + 2) < 0]The critical points are ( x = 0 ) and ( x = -2 ). Testing intervals around these points, we find the solution is ( -2 < x < 0 ).However, the problem asks for a sufficient but not necessary condition. This means we need an interval where if ( x ) is within it, ( f(x) < 1 ), but ( f(x) < 1 ) can also hold outside this interval. The interval ( -2 < x < 0 ) (option C) is the exact solution, making it necessary and sufficient. A subset of this interval, such as ( -1 < x < 0 ) (option B), is sufficient because it guarantees ( f(x) < 1 ), but not necessary because ( f(x) < 1 ) also holds for ( x ) in ( -2 < x < -1 ).Thus, the sufficient but not necessary condition is (boxed{B}).]

🔑:To find when f(x) 1, the function a^y is increasing for all real y. Therefore, we need x^2 + 2x < 0 for the inequality a^{x^{2}+2x} < 1 to be true.Now we factor the quadratic inequality:x(x + 2) < 0The roots of the equation x^2 + 2x = 0 are x = 0 and x = -2. Considering the sign change across the roots of a quadratic equation, we know that the inequality holds between the two roots, so we have:-2 < x < 0This is the necessary and sufficient condition for f(x) < 1.Now, we look for one condition that is sufficient but not necessary. This means that while it will ensure f(x) < 1, it does not cover all the cases. Among the options, -1 < x < 0 (Option B) is sufficient because it's a subset of the interval -2 < x < 0, but it's not necessary because not all values satisfying -2 < x < 0 are included.Therefore, the correct answer is (boxed{B}).