answer：First, I need to find the value of ( y ) using the given condition that the dot product of vectors ( overrightarrow{AB} ) and ( overrightarrow{AC} ) is zero.The dot product formula is:[overrightarrow{AB} cdot overrightarrow{AC} = (1)(2) + (y)(-1) = 0]Simplifying this equation:[2 - y = 0 implies y = 2]Now that I have ( y = 2 ), the vector ( overrightarrow{AB} ) becomes ( (1, 2) ).Next, I need to compute ( 3overrightarrow{AB} - 2overrightarrow{AC} ):[3overrightarrow{AB} = 3(1, 2) = (3, 6)][2overrightarrow{AC} = 2(2, -1) = (4, -2)]Subtracting these results:[3overrightarrow{AB} - 2overrightarrow{AC} = (3 - 4, 6 - (-2)) = (-1, 8)]Therefore, the final result is ( (-1, 8) ).

answer：Okay, so I have this problem about an ellipse, and I need to solve two parts. Let me start with part (1). The ellipse is given by the equation ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ) where ( a > b > 0 ). The right focus is at ( F(1, 0) ). I remember that for an ellipse, the distance from the center to each focus is ( c ), and ( c^2 = a^2 - b^2 ). Since the right focus is at (1, 0), that means ( c = 1 ). So, ( c = 1 ), which gives me ( c^2 = 1 ).The problem also mentions that when the line PQ passes through a vertex of the ellipse, its angle of inclination is exactly 60 degrees. Hmm, okay. So, the line PQ is passing through a vertex, which would be either (a, 0) or (-a, 0) since the major axis is along the x-axis because ( a > b ). Since the focus is at (1, 0), the center of the ellipse must be at (0, 0), right? Because the foci are symmetric about the center.So, the vertices are at (a, 0) and (-a, 0). The line PQ passes through one of these vertices, and the angle of inclination is 60 degrees. That means the slope of the line is ( tan(60^circ) = sqrt{3} ). So, the line PQ has a slope of ( sqrt{3} ) and passes through the vertex (a, 0) or (-a, 0). But since the line also passes through the focus F(1, 0), which is on the right side, I think the vertex it passes through is (a, 0). Because if it passes through (-a, 0), the line would have to go from (-a, 0) through (1, 0), but the angle of inclination is 60 degrees, which is positive, so it's probably going upwards from (a, 0).Wait, but actually, if the line passes through (a, 0) and has a slope of ( sqrt{3} ), it would go upwards to the right, but since the focus is at (1, 0), which is to the left of (a, 0) if a > 1. Hmm, maybe I need to think more carefully.Alternatively, maybe the line passes through the vertex (a, 0) and the focus (1, 0), and has a slope of ( sqrt{3} ). So, the line connects (a, 0) and (1, 0) with a slope of ( sqrt{3} ). But wait, the slope between (a, 0) and (1, 0) is zero because both are on the x-axis. That can't be. So, maybe it's not passing through (a, 0) but another vertex? Wait, the ellipse only has two vertices on the x-axis, at (a, 0) and (-a, 0). So, perhaps the line passes through (a, 0) and another point on the ellipse, making a 60-degree angle.Wait, maybe the line PQ is passing through the vertex (a, 0) and the focus (1, 0). But as I thought earlier, the slope between those two points is zero. So that can't be. Maybe the line is passing through another vertex, like the top vertex (0, b). But the angle of inclination is 60 degrees, so it's not vertical. Hmm, maybe I'm overcomplicating.Wait, the problem says "when the line PQ passes through a vertex of the ellipse, its angle of inclination is exactly 60 degrees." So, when PQ passes through a vertex, the angle is 60 degrees. So, PQ is a line passing through F(1, 0) and a vertex, making a 60-degree angle.So, the line passes through F(1, 0) and a vertex, say (a, 0). But then the slope would be zero, which is 0 degrees, not 60. So, maybe it's passing through (0, b). Let's check.If the line passes through F(1, 0) and (0, b), then the slope would be ( m = frac{b - 0}{0 - 1} = -b ). The angle of inclination is 60 degrees, so the slope should be ( tan(60^circ) = sqrt{3} ). But the slope here is -b, so ( -b = sqrt{3} ), which would mean ( b = -sqrt{3} ). But since b is positive, that can't be. Alternatively, if the angle is 60 degrees, the slope is ( sqrt{3} ), so maybe the line passes through (0, b) and F(1, 0), so the slope is ( frac{b - 0}{0 - 1} = -b ). So, ( -b = sqrt{3} ) implies ( b = -sqrt{3} ), which is not possible because b is positive. Alternatively, maybe the line passes through (0, -b), so the slope would be ( frac{-b - 0}{0 - 1} = frac{-b}{-1} = b ). So, if the slope is ( sqrt{3} ), then ( b = sqrt{3} ). That makes sense because b is positive.So, if the line passes through (0, -b) and F(1, 0), the slope is ( b ), which is ( sqrt{3} ). So, ( b = sqrt{3} ). Therefore, ( b^2 = 3 ). Since ( c = 1 ), and ( c^2 = a^2 - b^2 ), we have ( 1 = a^2 - 3 ), so ( a^2 = 4 ). Therefore, the equation of the ellipse is ( frac{x^2}{4} + frac{y^2}{3} = 1 ).Wait, but let me double-check. If the line passes through (0, -b) and F(1, 0), the slope is ( frac{0 - (-b)}{1 - 0} = frac{b}{1} = b ). The angle of inclination is 60 degrees, so the slope is ( tan(60^circ) = sqrt{3} ). Therefore, ( b = sqrt{3} ). So, yes, that's correct. Then ( a^2 = b^2 + c^2 = 3 + 1 = 4 ). So, the equation is ( frac{x^2}{4} + frac{y^2}{3} = 1 ).Okay, that seems solid. So, part (1) is done.Now, part (2). Let me read it again.Let O be the origin. Is there a point T(t, 0) on the line segment OF such that ( overrightarrow{QP} cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} )? If it exists, find the range of the real number t; if not, explain why.Hmm, okay. So, O is the origin, F is (1, 0), so the segment OF is from (0, 0) to (1, 0). So, T is somewhere on the x-axis between O and F, at (t, 0) where ( 0 leq t leq 1 ).We need to find if such a T exists such that the dot product ( overrightarrow{QP} cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} ).First, let me parse this. ( overrightarrow{QP} ) is the vector from Q to P, and ( overrightarrow{TP} ) is the vector from T to P. Similarly, ( overrightarrow{PQ} ) is the vector from P to Q, and ( overrightarrow{TQ} ) is the vector from T to Q.So, the equation is ( overrightarrow{QP} cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} ).Let me note that ( overrightarrow{QP} = -overrightarrow{PQ} ). So, ( overrightarrow{QP} = -overrightarrow{PQ} ). Therefore, the equation becomes:( -overrightarrow{PQ} cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} ).Let me factor out ( overrightarrow{PQ} ):( overrightarrow{PQ} cdot (-overrightarrow{TP} - overrightarrow{TQ}) = 0 ).Wait, no, let me think again. Let me write it step by step.Given:( overrightarrow{QP} cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} ).Replace ( overrightarrow{QP} ) with ( -overrightarrow{PQ} ):( (-overrightarrow{PQ}) cdot overrightarrow{TP} = overrightarrow{PQ} cdot overrightarrow{TQ} ).Bring all terms to one side:( (-overrightarrow{PQ}) cdot overrightarrow{TP} - overrightarrow{PQ} cdot overrightarrow{TQ} = 0 ).Factor out ( overrightarrow{PQ} ):( overrightarrow{PQ} cdot (-overrightarrow{TP} - overrightarrow{TQ}) = 0 ).Simplify the vectors inside the dot product:( -overrightarrow{TP} - overrightarrow{TQ} = -(overrightarrow{TP} + overrightarrow{TQ}) ).But ( overrightarrow{TP} + overrightarrow{TQ} = overrightarrow{P} - overrightarrow{T} + overrightarrow{Q} - overrightarrow{T} = (overrightarrow{P} + overrightarrow{Q}) - 2overrightarrow{T} ).Wait, maybe another approach. Let me think in terms of coordinates.Let me denote points P and Q as ( P(x_1, y_1) ) and ( Q(x_2, y_2) ). Then, vectors:( overrightarrow{QP} = (x_1 - x_2, y_1 - y_2) ).( overrightarrow{TP} = (x_1 - t, y_1 - 0) = (x_1 - t, y_1) ).Similarly,( overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) = -overrightarrow{QP} ).( overrightarrow{TQ} = (x_2 - t, y_2) ).So, the equation is:( (x_1 - x_2)(x_1 - t) + (y_1 - y_2)y_1 = (x_2 - x_1)(x_2 - t) + (y_2 - y_1)y_2 ).Let me write this out:Left side: ( (x_1 - x_2)(x_1 - t) + (y_1 - y_2)y_1 ).Right side: ( (x_2 - x_1)(x_2 - t) + (y_2 - y_1)y_2 ).Let me expand both sides.Left side:( (x_1 - x_2)(x_1 - t) = x_1^2 - t x_1 - x_1 x_2 + t x_2 ).( (y_1 - y_2)y_1 = y_1^2 - y_1 y_2 ).So, left side total:( x_1^2 - t x_1 - x_1 x_2 + t x_2 + y_1^2 - y_1 y_2 ).Right side:( (x_2 - x_1)(x_2 - t) = x_2^2 - t x_2 - x_1 x_2 + t x_1 ).( (y_2 - y_1)y_2 = y_2^2 - y_1 y_2 ).So, right side total:( x_2^2 - t x_2 - x_1 x_2 + t x_1 + y_2^2 - y_1 y_2 ).Now, set left side equal to right side:( x_1^2 - t x_1 - x_1 x_2 + t x_2 + y_1^2 - y_1 y_2 = x_2^2 - t x_2 - x_1 x_2 + t x_1 + y_2^2 - y_1 y_2 ).Let me subtract the right side from both sides to bring everything to the left:( x_1^2 - t x_1 - x_1 x_2 + t x_2 + y_1^2 - y_1 y_2 - x_2^2 + t x_2 + x_1 x_2 - t x_1 - y_2^2 + y_1 y_2 = 0 ).Simplify term by term:- ( x_1^2 - x_2^2 )- ( -t x_1 + t x_2 + t x_2 - t x_1 ) → ( -2 t x_1 + 2 t x_2 )- ( -x_1 x_2 + x_1 x_2 ) → 0- ( y_1^2 - y_2^2 )- ( - y_1 y_2 + y_1 y_2 ) → 0So, the equation simplifies to:( (x_1^2 - x_2^2) + (-2 t x_1 + 2 t x_2) + (y_1^2 - y_2^2) = 0 ).Factor the differences of squares:( (x_1 - x_2)(x_1 + x_2) + 2 t (x_2 - x_1) + (y_1 - y_2)(y_1 + y_2) = 0 ).Factor out ( (x_1 - x_2) ) and ( (y_1 - y_2) ):( (x_1 - x_2)(x_1 + x_2 - 2 t) + (y_1 - y_2)(y_1 + y_2) = 0 ).Hmm, interesting. Let me denote ( x_1 + x_2 = S_x ), ( y_1 + y_2 = S_y ), and ( x_1 - x_2 = D_x ), ( y_1 - y_2 = D_y ). Then, the equation becomes:( D_x (S_x - 2 t) + D_y S_y = 0 ).But I'm not sure if that helps. Maybe another approach.Since P and Q lie on the ellipse, they satisfy ( frac{x_1^2}{4} + frac{y_1^2}{3} = 1 ) and ( frac{x_2^2}{4} + frac{y_2^2}{3} = 1 ). So, subtracting these two equations:( frac{x_1^2 - x_2^2}{4} + frac{y_1^2 - y_2^2}{3} = 0 ).Which can be written as:( frac{(x_1 - x_2)(x_1 + x_2)}{4} + frac{(y_1 - y_2)(y_1 + y_2)}{3} = 0 ).Let me denote ( D_x = x_1 - x_2 ), ( S_x = x_1 + x_2 ), ( D_y = y_1 - y_2 ), ( S_y = y_1 + y_2 ). Then, the above equation becomes:( frac{D_x S_x}{4} + frac{D_y S_y}{3} = 0 ).So, ( frac{D_x S_x}{4} = - frac{D_y S_y}{3} ).Let me write this as:( frac{D_x}{D_y} = - frac{4 S_y}{3 S_x} ).But from the earlier equation we had:( D_x (S_x - 2 t) + D_y S_y = 0 ).Let me substitute ( D_x = - frac{4 S_y}{3 S_x} D_y ) into this equation.So:( - frac{4 S_y}{3 S_x} D_y (S_x - 2 t) + D_y S_y = 0 ).Factor out ( D_y S_y ):( D_y S_y left( - frac{4}{3 S_x} (S_x - 2 t) + 1 right) = 0 ).Assuming ( D_y neq 0 ) and ( S_y neq 0 ) (since P and Q are distinct points and the line isn't horizontal), we can divide both sides by ( D_y S_y ):( - frac{4}{3 S_x} (S_x - 2 t) + 1 = 0 ).Simplify:( - frac{4}{3} + frac{8 t}{3 S_x} + 1 = 0 ).Combine constants:( - frac{4}{3} + 1 = - frac{1}{3} ).So:( - frac{1}{3} + frac{8 t}{3 S_x} = 0 ).Multiply both sides by 3:( -1 + frac{8 t}{S_x} = 0 ).Thus:( frac{8 t}{S_x} = 1 ).So,( S_x = 8 t ).But ( S_x = x_1 + x_2 ). So, ( x_1 + x_2 = 8 t ).Hmm, interesting. So, the sum of the x-coordinates of P and Q is 8t.But I also know that the line PQ passes through F(1, 0), so the equation of PQ can be written as ( y = k(x - 1) ), where k is the slope.Let me find the points P and Q by solving the system of the ellipse and the line.Substitute ( y = k(x - 1) ) into the ellipse equation:( frac{x^2}{4} + frac{(k(x - 1))^2}{3} = 1 ).Expand this:( frac{x^2}{4} + frac{k^2 (x^2 - 2x + 1)}{3} = 1 ).Multiply both sides by 12 to eliminate denominators:( 3 x^2 + 4 k^2 (x^2 - 2x + 1) = 12 ).Expand:( 3 x^2 + 4 k^2 x^2 - 8 k^2 x + 4 k^2 = 12 ).Combine like terms:( (3 + 4 k^2) x^2 - 8 k^2 x + (4 k^2 - 12) = 0 ).This is a quadratic in x. Let me denote this as:( A x^2 + B x + C = 0 ),where ( A = 3 + 4 k^2 ),( B = -8 k^2 ),( C = 4 k^2 - 12 ).The solutions are x-coordinates of P and Q. Let me denote them as ( x_1 ) and ( x_2 ). Then, from Vieta's formulas:( x_1 + x_2 = -B/A = 8 k^2 / (3 + 4 k^2) ).But earlier, we have ( x_1 + x_2 = 8 t ). Therefore,( 8 k^2 / (3 + 4 k^2) = 8 t ).Divide both sides by 8:( k^2 / (3 + 4 k^2) = t ).So,( t = frac{k^2}{3 + 4 k^2} ).Hmm, so t is expressed in terms of k. Since k is the slope of the line PQ, which is not perpendicular to the coordinate axes, so k ≠ 0 and k is real.We need to find the range of t. Since k^2 is positive, let me denote ( m = k^2 ), so ( m > 0 ).Then,( t = frac{m}{3 + 4 m} ).We can write this as:( t = frac{1}{(3/m) + 4} ).As m varies from 0 to infinity, ( 3/m ) varies from infinity to 0. Therefore, ( (3/m) + 4 ) varies from infinity to 4. Hence, ( t = 1 / (something ) between 4 and infinity) varies from 0 to 1/4.Therefore, t ∈ (0, 1/4).So, there exists such a point T(t, 0) on segment OF, and t must be between 0 and 1/4.Wait, but let me check. If m approaches 0, t approaches 0. If m approaches infinity, t approaches ( frac{m}{4 m} = 1/4 ). So, yes, t ∈ (0, 1/4).Therefore, the answer is that such a point T exists, and t is in (0, 1/4).I think that's it.**Final Answer**(1) The equation of the ellipse is (boxed{dfrac{x^2}{4} + dfrac{y^2}{3} = 1}).(2) The range of the real number ( t ) is (boxed{left(0, dfrac{1}{4}right)}).

answer：Okay, so I have this problem about a rectangle ABCD where AB is 10 cm and BC is 12 cm. They mention points E and F such that DE equals DF, and the area of triangle DEF is one-third the area of the rectangle. I need to find the length of segment EF. Hmm, let me try to visualize this.First, let me sketch the rectangle in my mind. Rectangle ABCD means that AB and CD are the lengths, each 10 cm, and BC and AD are the widths, each 12 cm. So, it's a rectangle that's taller than it is wide. Now, points E and F are somewhere on the rectangle, but the problem doesn't specify where exactly. It just says DE equals DF, so triangle DEF is an isosceles triangle with DE and DF as the equal sides.The area of the rectangle is straightforward to calculate. Since area of a rectangle is length times width, that would be 10 cm multiplied by 12 cm, which is 120 cm². The area of triangle DEF is one-third of that, so that's 120 divided by 3, which is 40 cm². Okay, so triangle DEF has an area of 40 cm².Now, since DE equals DF, triangle DEF is isosceles, and the area can be calculated using the formula for the area of a triangle: (base * height)/2. But in this case, since two sides are equal, maybe I can express the area in terms of DE and DF. Wait, but DE and DF are equal, so maybe I can use the formula for the area of an isosceles triangle.Alternatively, since DE and DF are equal, maybe I can consider them as the legs of a right triangle, but I don't know if DEF is a right triangle. Hmm, the problem doesn't specify that, so I can't assume that. Maybe I need to use coordinates to figure this out.Let me assign coordinates to the rectangle. Let's place point D at the origin (0,0) for simplicity. Then, since AB is 10 cm and BC is 12 cm, the rectangle would have points at D(0,0), C(10,0), B(10,12), and A(0,12). So, points E and F must be somewhere on the sides of the rectangle.Wait, but the problem doesn't specify where E and F are located. They could be on any of the sides. Hmm, this is a bit confusing. Maybe I need to make some assumptions here. Since DE and DF are equal, points E and F must be equidistant from D. So, if D is at (0,0), then E and F must lie on a circle centered at D with radius DE (which is equal to DF). But since E and F are points on the rectangle, they must lie on the sides of the rectangle. So, E and F are points on the sides of the rectangle such that their distance from D is equal. Let me think about where these points could be.If I consider the sides adjacent to D, which are DC and DA. So, point E could be on DC and point F on DA, or both on the same side, but since DE and DF are equal, they might be on different sides. Let me try to figure this out.Suppose E is on side DC and F is on side DA. Then, the coordinates of E would be (x,0) and F would be (0,y), where x and y are between 0 and 10 and 0 and 12, respectively. Since DE = DF, the distance from D(0,0) to E(x,0) is x, and the distance from D(0,0) to F(0,y) is y. Therefore, x must equal y. So, E is (x,0) and F is (0,x), since x = y.Now, the area of triangle DEF can be calculated using the coordinates. The area can be found using the determinant formula:Area = (1/2) | (x * x - 0 * 0) | = (1/2) x².Wait, that doesn't seem right. Let me recall the formula for the area of a triangle given three points. If I have points D(0,0), E(x,0), and F(0,x), then the area is (1/2)*base*height. Here, the base can be DE, which is x, and the height can be DF, which is x, but since they are perpendicular, the area is (1/2)*x*x = (1/2)x².We know the area is 40 cm², so:(1/2)x² = 40 x² = 80 x = sqrt(80) = 4*sqrt(5)Wait, but hold on. If E is on DC and F is on DA, then the coordinates are (x,0) and (0,x). Then, the distance EF would be the distance between (x,0) and (0,x), which is sqrt((x-0)² + (0 - x)²) = sqrt(x² + x²) = sqrt(2x²) = x*sqrt(2).Since x is 4*sqrt(5), then EF would be 4*sqrt(5)*sqrt(2) = 4*sqrt(10). Hmm, that seems plausible.But wait, let me double-check my assumption. I assumed that E is on DC and F is on DA. Is there another possibility? For example, could E and F both be on side DC or DA? If both E and F are on DC, then DE and DF would both be along DC, but then triangle DEF would collapse into a line, which doesn't make sense. Similarly, if both are on DA, the same issue arises. So, E and F must be on different sides.Alternatively, E could be on DC and F on AB, but then DF would be longer than DE, unless F is on DA. Wait, no, because if F is on AB, then DF would be the distance from D(0,0) to F on AB, which is at (something,12). That distance would be sqrt((something)^2 + 12^2), which is definitely longer than DE, which is along DC. So, unless E is somewhere else, but the problem states DE = DF, so they must be equal.Therefore, my initial assumption that E is on DC and F is on DA seems correct because that allows DE and DF to be equal, both being x, as I set earlier.So, with that, I found that x is 4*sqrt(5), so EF is 4*sqrt(10). Let me just confirm the area calculation again.Area of triangle DEF is (1/2)*x² = 40. So, x² = 80, x = sqrt(80) = 4*sqrt(5). Then, EF is the distance between (x,0) and (0,x), which is sqrt(x² + x²) = sqrt(2x²) = x*sqrt(2) = 4*sqrt(5)*sqrt(2) = 4*sqrt(10). Yeah, that seems consistent.Wait, but hold on. Is triangle DEF necessarily a right triangle? Because in my calculation, I assumed that the area is (1/2)*x², which is only true if the triangle is right-angled at D. But the problem doesn't specify that DEF is a right triangle. Hmm, that might be an oversight.So, maybe I need to approach this differently. Instead of assuming it's a right triangle, I should use the general formula for the area of a triangle given two sides and the included angle. The area is (1/2)*DE*DF*sin(theta), where theta is the angle between DE and DF.Since DE = DF = x, the area becomes (1/2)*x²*sin(theta) = 40. So, x²*sin(theta) = 80.But without knowing theta, I can't directly find x. Hmm, this complicates things. Maybe I need another approach.Alternatively, perhaps using coordinates is still the way to go, but without assuming the triangle is right-angled. Let me try that.Let me assign coordinates again: D(0,0), C(10,0), B(10,12), A(0,12). Let E be a point on side DC, so E has coordinates (e,0), where 0 ≤ e ≤ 10. Let F be a point on side DA, so F has coordinates (0,f), where 0 ≤ f ≤ 12.Given that DE = DF, the distance from D to E is e, and the distance from D to F is f. So, e = f.Therefore, E is (e,0) and F is (0,e), since e = f.Now, the area of triangle DEF can be calculated using the determinant formula:Area = (1/2)| (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) |.Plugging in the coordinates D(0,0), E(e,0), F(0,e):Area = (1/2)| 0*(0 - e) + e*(e - 0) + 0*(0 - 0) | = (1/2)| 0 + e² + 0 | = (1/2)e².We know the area is 40, so:(1/2)e² = 40 e² = 80 e = sqrt(80) = 4*sqrt(5).So, E is (4√5, 0) and F is (0,4√5). Then, the distance EF is the distance between these two points:EF = sqrt[(4√5 - 0)² + (0 - 4√5)²] = sqrt[(16*5) + (16*5)] = sqrt[80 + 80] = sqrt[160] = 4*sqrt(10).Okay, so even when considering the general case without assuming the triangle is right-angled, I still arrive at the same result. Therefore, EF is 4√10 cm.Wait, but just to be thorough, could E and F be on other sides? For example, E on DC and F on AB? Let's explore that possibility.If E is on DC at (e,0) and F is on AB at (f,12), then DE is e and DF is sqrt(f² + 12²). Since DE = DF, we have:e = sqrt(f² + 144).But then, the area of triangle DEF would be more complicated to calculate. Let me see.Using coordinates D(0,0), E(e,0), F(f,12). The area can be calculated using the determinant formula:Area = (1/2)| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) | = (1/2)| 0*(0 - 12) + e*(12 - 0) + f*(0 - 0) | = (1/2)| 0 + 12e + 0 | = (1/2)(12e) = 6e.We know the area is 40, so:6e = 40 e = 40/6 ≈ 6.666...But from DE = DF, we have e = sqrt(f² + 144). So,sqrt(f² + 144) = 40/6 ≈ 6.666...But sqrt(f² + 144) must be at least 12, since f² is non-negative. However, 40/6 is approximately 6.666, which is less than 12. This is impossible because sqrt(f² + 144) ≥ 12. Therefore, this scenario is not possible.So, E cannot be on DC and F on AB because it leads to a contradiction. Similarly, if E is on DA and F on BC, we might run into similar issues.Alternatively, if E is on BC and F is on AD, let's see. Let me assign coordinates accordingly. Let E be on BC at (10, e) and F be on AD at (0, f). Then, DE is the distance from D(0,0) to E(10,e), which is sqrt(10² + e²). Similarly, DF is the distance from D(0,0) to F(0,f), which is f. Since DE = DF, we have:sqrt(100 + e²) = f.The area of triangle DEF can be calculated using the determinant formula:Area = (1/2)| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) | = (1/2)| 0*(e - f) + 10*(f - 0) + 0*(0 - e) | = (1/2)| 0 + 10f + 0 | = (1/2)(10f) = 5f.We know the area is 40, so:5f = 40 f = 8.Then, from DE = DF, we have sqrt(100 + e²) = 8. But sqrt(100 + e²) must be at least 10, since e² is non-negative. However, 8 is less than 10, which is impossible. Therefore, this scenario is also not possible.So, the only feasible scenario is when E is on DC and F is on DA, both at a distance of 4√5 from D. Therefore, the length of EF is 4√10 cm.Just to recap, the steps I took were:1. Calculate the area of the rectangle.2. Determine the area of triangle DEF.3. Assign coordinates to the rectangle and points E and F.4. Use the condition DE = DF to relate the coordinates.5. Calculate the area of triangle DEF using coordinates and set it equal to 40 cm².6. Solve for the coordinates and then compute the distance EF.7. Consider alternative positions for E and F but find them impossible due to contradictions.8. Conclude that the initial assumption was correct and compute EF accordingly.This seems thorough, and I don't see any mistakes in the logic or calculations. So, I'm confident that the length of segment EF is 4√10 centimeters.

answer：Okay, so I have this problem where I need to find the sum of a series that starts with -1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + ... and continues up to 14400. The signs change every (5k)^2 for k in natural numbers. Hmm, that means the sign changes happen at 25, 100, 225, and so on, right?Let me try to understand the pattern here. The first term is -1, then the next three terms are positive: +2, +3, +4. Then the next five terms are negative: -5, -6, -7, -8, -9. I wonder if this pattern continues? So, the number of terms with the same sign increases by two each time? Let me check:- First block: 1 term (negative)- Second block: 3 terms (positive)- Third block: 5 terms (negative)- Fourth block: 7 terms (positive)- And so on...Wait, that seems to be the case. Each block has an odd number of terms, increasing by two each time, and the signs alternate starting with negative. So, the first block is negative, the second is positive, the third is negative, etc.Now, the problem mentions that the signs change every (5k)^2. Let me verify this. The first sign change is at 25, which is (5*1)^2. The next sign change is at 100, which is (5*2)^2. Then 225, which is (5*3)^2, and so on. So, each time k increases by 1, the sign changes at (5k)^2.So, the blocks are defined between these squares. That is, the first block is from 1 to 25, the second from 26 to 100, the third from 101 to 225, and so on. Each block has a length of (5k)^2 - (5(k-1))^2. Let me compute that:(5k)^2 - (5(k-1))^2 = 25k² - 25(k² - 2k + 1) = 25k² - 25k² + 50k -25 = 50k -25.So, each block has 50k -25 terms. But wait, earlier I thought the number of terms in each block increases by two each time. Let me see:For k=1: 50*1 -25 =25 terms. But the first block is only 1 term. Hmm, that doesn't match. Maybe my initial assumption about the blocks is wrong.Wait, perhaps the sign changes at (5k)^2, meaning that the sign for the block starting at (5(k-1))² +1 up to (5k)^2 is determined by k. Let me try that.So, for k=1: from 1 to 25, sign is negative.For k=2: from 26 to 100, sign is positive.For k=3: from 101 to 225, sign is negative.And so on.So, each block from (5(k-1))² +1 to (5k)^2 has a sign of (-1)^k.Wait, let me check:k=1: (-1)^1 = -1, so negative. That's correct.k=2: (-1)^2 = +1, positive. Correct.k=3: (-1)^3 = -1, negative. Correct.So, the sign for each block is (-1)^k, and the number of terms in each block is (5k)^2 - (5(k-1))^2 = 50k -25, as I calculated earlier.So, the sum can be written as the sum over k from 1 to some maximum k, of (-1)^k multiplied by the sum of numbers from (5(k-1))² +1 to (5k)^2.Let me denote S as the total sum. Then,S = Σ [ (-1)^k * Σ (n) ] from n = (5(k-1))² +1 to (5k)^2, for k from 1 to K, where K is such that (5K)^2 =14400.Wait, 14400 is (120)^2, so 5K=120, so K=24.So, K=24. So, we need to sum from k=1 to k=24.So, S = Σ [ (-1)^k * Σ (n) ] from n=(5(k-1))² +1 to (5k)^2, for k=1 to 24.Now, the inner sum is the sum of consecutive numbers from a to b, which is (b - a +1)(a + b)/2.So, let's compute that.For each k, a = (5(k-1))² +1, b= (5k)^2.So, the number of terms is b - a +1 = (5k)^2 - (5(k-1))² = 50k -25, as before.The sum is (number of terms)*(a + b)/2.So, sum = (50k -25)*( ( (5(k-1))² +1 + (5k)^2 ) / 2 )Let me compute (5(k-1))² +1 + (5k)^2:= 25(k-1)^2 +1 +25k²= 25(k² - 2k +1) +1 +25k²=25k² -50k +25 +1 +25k²=50k² -50k +26So, the sum is (50k -25)*(50k² -50k +26)/2Simplify:= (50k -25)*(50k² -50k +26)/2= (25*(2k -1))*(50k² -50k +26)/2=25*(2k -1)*(50k² -50k +26)/2Hmm, that's a bit messy. Maybe I can factor further or simplify.Alternatively, perhaps I can write the entire expression for S as:S = Σ [ (-1)^k * (50k -25)*(50k² -50k +26)/2 ] from k=1 to 24.This seems complicated, but maybe I can expand the terms and then sum them up.Let me try to expand the expression inside the sum:(50k -25)*(50k² -50k +26) = 50k*(50k² -50k +26) -25*(50k² -50k +26)=2500k³ -2500k² +1300k -1250k² +1250k -650=2500k³ - (2500 +1250)k² + (1300 +1250)k -650=2500k³ -3750k² +2550k -650So, the sum becomes:S = Σ [ (-1)^k * (2500k³ -3750k² +2550k -650)/2 ] from k=1 to24.Simplify by dividing each term by 2:= Σ [ (-1)^k * (1250k³ -1875k² +1275k -325) ] from k=1 to24.So, S = Σ [ (-1)^k * (1250k³ -1875k² +1275k -325) ] from k=1 to24.Now, this is a sum of terms with alternating signs. So, we can write it as:S = Σ [ (-1)^k *1250k³ ] - Σ [ (-1)^k *1875k² ] + Σ [ (-1)^k *1275k ] - Σ [ (-1)^k *325 ] from k=1 to24.Factor out the constants:=1250 Σ [ (-1)^k k³ ] -1875 Σ [ (-1)^k k² ] +1275 Σ [ (-1)^k k ] -325 Σ [ (-1)^k ] from k=1 to24.Now, we need to compute these four sums separately.Let me denote:A = Σ [ (-1)^k k³ ] from k=1 to24.B = Σ [ (-1)^k k² ] from k=1 to24.C = Σ [ (-1)^k k ] from k=1 to24.D = Σ [ (-1)^k ] from k=1 to24.So, S =1250A -1875B +1275C -325D.Now, let's compute A, B, C, D.Starting with D:D = Σ [ (-1)^k ] from k=1 to24.This is an alternating sum of 1 and -1, 24 terms.Since 24 is even, the sum is 0. Because each pair (-1 +1) cancels out.So, D=0.Next, C = Σ [ (-1)^k k ] from k=1 to24.This is the alternating sum of k from 1 to24.We can write this as:C = (-1 +2) + (-3 +4) + ... + (-23 +24).Each pair is ( - (2m-1) + 2m ) =1, for m=1 to12.So, there are 12 pairs, each summing to1. So, C=12.Wait, let me verify:From k=1 to24:(-1) +2 + (-3)+4 +...+(-23)+24.Grouping as (-1+2) + (-3+4)+...+(-23+24). Each pair is 1, and there are12 pairs. So, C=12.Good.Now, B = Σ [ (-1)^k k² ] from k=1 to24.This is more complicated. Let's see if we can find a pattern or formula.I recall that Σ [ (-1)^k k² ] from k=1 to n can be expressed in terms of n.Let me try to compute it for n=24.Alternatively, maybe I can find a recurrence or use known summation formulas.Alternatively, perhaps compute it by grouping terms.Let me write out the sum:B = (-1)^1*1² + (-1)^2*2² + (-1)^3*3² + ... + (-1)^24*24².Which is:-1 +4 -9 +16 -25 +36 -49 +64 -81 +100 -121 +144 -169 +196 -225 +256 -289 +324 -361 +400 -441 +484 -529 +576.Hmm, that's a lot of terms. Maybe I can group them in pairs:(-1 +4) + (-9 +16) + (-25 +36) + (-49 +64) + (-81 +100) + (-121 +144) + (-169 +196) + (-225 +256) + (-289 +324) + (-361 +400) + (-441 +484) + (-529 +576).Each pair is ( - (2m-1)^2 + (2m)^2 ) for m=1 to12.Compute each pair:( - (2m-1)^2 + (2m)^2 ) = - (4m² -4m +1) +4m² = -4m² +4m -1 +4m² =4m -1.So, each pair sums to4m -1.So, B = Σ [4m -1] from m=1 to12.Compute this sum:Σ [4m -1] =4 Σ m - Σ1 =4*(12*13/2) -12 =4*78 -12=312 -12=300.So, B=300.Wait, let me verify:Each pair is ( - (2m-1)^2 + (2m)^2 )=4m -1.So, summing from m=1 to12: Σ(4m -1)=4*(1+2+...+12) -12=4*(78) -12=312 -12=300. Yes, correct.So, B=300.Now, A = Σ [ (-1)^k k³ ] from k=1 to24.This is even more complicated. Let me see if I can find a pattern or formula.I know that Σ [ (-1)^k k³ ] can be expressed in terms of n, but I don't remember the exact formula. Maybe I can derive it.Alternatively, perhaps compute it by grouping terms.Let me try grouping terms in pairs:A = (-1)^1*1³ + (-1)^2*2³ + (-1)^3*3³ + ... + (-1)^24*24³.Which is:-1 +8 -27 +64 -125 +216 -343 +512 -729 +1000 -1331 +1728 -2197 +2744 -3375 +4096 -4913 +5832 -6859 +8000 -9261 +10648 -12167 +13824.Again, grouping in pairs:(-1 +8) + (-27 +64) + (-125 +216) + (-343 +512) + (-729 +1000) + (-1331 +1728) + (-2197 +2744) + (-3375 +4096) + (-4913 +5832) + (-6859 +8000) + (-9261 +10648) + (-12167 +13824).Each pair is ( - (2m-1)^3 + (2m)^3 ) for m=1 to12.Compute each pair:( - (2m-1)^3 + (2m)^3 ) = - (8m³ -12m² +6m -1) +8m³ = -8m³ +12m² -6m +1 +8m³ =12m² -6m +1.So, each pair sums to12m² -6m +1.Therefore, A = Σ [12m² -6m +1] from m=1 to12.Compute this sum:Σ [12m² -6m +1] =12 Σ m² -6 Σ m + Σ1.Compute each term:Σ m² from1 to12= (12)(13)(25)/6= (12*13*25)/6= (2*13*25)=650.Σ m from1 to12=78.Σ1 from1 to12=12.So,12*650=7800-6*78= -468+12= +12Total=7800 -468 +12=7800 -456=7344.So, A=7344.Wait, let me verify:Each pair is12m² -6m +1.Sum from m=1 to12:12*(1² +2² +...+12²) -6*(1+2+...+12) +12*1.We know Σm²=650, Σm=78.So,12*650=7800-6*78= -468+12= +12Total=7800 -468 +12=7344. Correct.So, A=7344.Now, putting it all together:S=1250A -1875B +1275C -325D.We have:A=7344B=300C=12D=0So,S=1250*7344 -1875*300 +1275*12 -325*0.Compute each term:1250*7344:First, 1250*7000=8,750,0001250*344=1250*(300+44)=1250*300=375,000 +1250*44=55,000Total=375,000 +55,000=430,000So, total 1250*7344=8,750,000 +430,000=9,180,000.Next, -1875*300= -562,500.Then, 1275*12=15,300.Finally, -325*0=0.So, S=9,180,000 -562,500 +15,300.Compute step by step:9,180,000 -562,500=8,617,5008,617,500 +15,300=8,632,800.So, S=8,632,800.Wait, that seems quite large. Let me check my calculations again.First, A=7344, B=300, C=12, D=0.Compute 1250*A=1250*7344.Let me compute 7344*1000=7,344,0007344*250=7344*200 +7344*50=1,468,800 +367,200=1,836,000So, 7344*1250=7,344,000 +1,836,000=9,180,000. Correct.Next, -1875*B= -1875*300= -562,500. Correct.1275*C=1275*12=15,300. Correct.So, total S=9,180,000 -562,500 +15,300=9,180,000 -562,500=8,617,500 +15,300=8,632,800.Hmm, that seems correct, but let me think about the initial terms.Wait, the first block is from1 to25, which is25 terms, all negative. Then the next block is26 to100, which is75 terms, positive. Then101 to225, which is125 terms, negative, and so on.Wait, but when I computed the sum for each block, I used the formula for the sum of consecutive numbers, which is correct. Then I expanded the expression and got A=7344, which seems correct given the steps.But let me think about the overall approach. Is there a simpler way to compute this sum?Alternatively, perhaps I made a mistake in the expansion of the terms.Wait, let me go back to the expression:sum = (50k -25)*(50k² -50k +26)/2I expanded this as:(50k -25)*(50k² -50k +26)=2500k³ -3750k² +2550k -650.Wait, let me verify this multiplication:(50k -25)*(50k² -50k +26)=50k*(50k² -50k +26) -25*(50k² -50k +26)=2500k³ -2500k² +1300k -1250k² +1250k -650Now, combine like terms:2500k³-2500k² -1250k²= -3750k²1300k +1250k=2550k-650Yes, correct.So, the expansion is correct.Then, dividing by2:(2500k³ -3750k² +2550k -650)/2=1250k³ -1875k² +1275k -325.So, that's correct.Then, S=Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ] from1 to24.Which is1250A -1875B +1275C -325D.With A=7344, B=300, C=12, D=0.So, S=1250*7344 -1875*300 +1275*12=9,180,000 -562,500 +15,300=8,632,800.Hmm, that seems correct, but let me think about the magnitude. The series goes up to14400, so the total number of terms is14400, and each term is up to14400. So, the sum could indeed be in the millions.But let me check if I made a mistake in the sign when computing A, B, C, D.Wait, in the expression S=1250A -1875B +1275C -325D.But A=Σ [ (-1)^k k³ ] from1 to24.Which was computed as7344. Let me check if that's correct.Wait, when I computed A=Σ [ (-1)^k k³ ] from1 to24, I grouped terms in pairs and found that each pair sums to12m² -6m +1, and then summed that from m=1 to12 to get7344.But let me think about the sign. The first term is (-1)^1*1³=-1, then (-1)^2*2³=+8, etc.When I grouped them as (-1 +8) + (-27 +64)+..., each pair is ( - (2m-1)^3 + (2m)^3 ). So, each pair is positive because (2m)^3 > (2m-1)^3.So, each pair is positive, and their sum is12m² -6m +1, which is positive.So, A is the sum of these positive pairs, which is7344.Similarly, B=Σ [ (-1)^k k² ] from1 to24=300, which is positive because each pair sums to4m -1, which is positive.C=12, positive.D=0.So, S=1250*7344 -1875*300 +1275*12.Which is positive, as computed.So, the final answer is8,632,800.Wait, but let me think again. The first block is negative, so the first term is negative, but the sum of the first block is negative, the second block is positive, etc.But when I computed A=Σ [ (-1)^k k³ ] from1 to24=7344, which is positive because the sum of positive pairs.Wait, but in reality, the first term is negative, so the first pair is (-1 +8)=7, which is positive, then (-27 +64)=37, positive, etc. So, each pair is positive, so A is positive.Similarly, B=300, positive.C=12, positive.So, S=1250*7344 -1875*300 +1275*12=8,632,800.I think that's correct.But let me check with a smaller case to see if the approach is correct.Suppose we take k=1 and k=2, so up to (5*2)^2=100.Compute the sum manually and see if it matches the formula.Compute S manually:From1 to25: negative.Sum1= - (1+2+...+25)= - (25*26)/2= -325.From26 to100: positive.Sum2=26+27+...+100= (100*101)/2 - (25*26)/2=5050 -325=4725.Total sum up to100: -325 +4725=4400.Now, using the formula:For k=1: (-1)^1*(1250*1³ -1875*1² +1275*1 -325)= -1250 -1875 +1275 -325= -1250 -1875= -3125 +1275= -1850 -325= -2175.For k=2: (-1)^2*(1250*8 -1875*4 +1275*2 -325)=1250*8=10,000 -1875*4=7,500 +1275*2=2,550 -325.So, 10,000 -7,500=2,500 +2,550=5,050 -325=4,725.So, total S= -2175 +4,725=2,550.Wait, but manually, we got4400. So, discrepancy here.Wait, that means my formula is incorrect.Wait, what's happening.Wait, in the formula, I have S=Σ [ (-1)^k * (1250k³ -1875k² +1275k -325) ] from k=1 to24.But when I compute for k=1 and k=2, I get S= -2175 +4,725=2,550, but manually, it's4400.So, clearly, my formula is wrong.Wait, where did I go wrong?Let me go back.I think the mistake is in the initial expression for the inner sum.I had:sum = (50k -25)*(50k² -50k +26)/2.But let me re-examine that.Wait, the inner sum is from a=(5(k-1))² +1 to b=(5k)^2.The number of terms is b -a +1= (5k)^2 - (5(k-1))²=50k -25.The sum is (number of terms)*(a + b)/2.So, a=(5(k-1))² +1=25(k-1)^2 +1.b=(5k)^2=25k².So, a + b=25(k-1)^2 +1 +25k²=25(k² -2k +1) +1 +25k²=25k² -50k +25 +1 +25k²=50k² -50k +26.So, sum=(50k -25)*(50k² -50k +26)/2.Yes, that's correct.Then, expanding:(50k -25)*(50k² -50k +26)=2500k³ -2500k² +1300k -1250k² +1250k -650=2500k³ -3750k² +2550k -650.Divide by2:1250k³ -1875k² +1275k -325.So, that's correct.But when I plug in k=1 and k=2, the formula doesn't match the manual sum.Wait, for k=1:sum= (50*1 -25)*(50*1² -50*1 +26)/2=(25)*(50 -50 +26)/2=25*26/2=25*13=325.But with the sign (-1)^1, it's -325, which matches the manual sum.Similarly, for k=2:sum=(50*2 -25)*(50*4 -50*2 +26)/2=(75)*(200 -100 +26)/2=75*126/2=75*63=4725.With sign (-1)^2=+1, so +4725.Total sum up to k=2: -325 +4725=4400, which matches manual.But according to my formula, S=Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ].For k=1: (-1)^1*(1250 -1875 +1275 -325)= -1250 -1875 +1275 -325= -1250 -1875= -3125 +1275= -1850 -325= -2175.But the actual sum for k=1 is -325.Wait, so my formula is incorrect.Wait, I think I made a mistake in the expansion.Wait, when I expanded (50k -25)*(50k² -50k +26), I got2500k³ -3750k² +2550k -650.But let me recompute:(50k -25)*(50k² -50k +26)=50k*(50k²) +50k*(-50k) +50k*26 -25*(50k²) -25*(-50k) -25*26=2500k³ -2500k² +1300k -1250k² +1250k -650.Yes, that's correct.So, 2500k³ -2500k² -1250k²=2500k³ -3750k².1300k +1250k=2550k.-650.So, correct.Then, dividing by2:1250k³ -1875k² +1275k -325.So, correct.But when k=1, the inner sum is325, but according to the formula, it's1250 -1875 +1275 -325= (1250 +1275) - (1875 +325)=2525 -2200=325.Wait, but with the sign (-1)^k, for k=1, it's -325, which is correct.Similarly, for k=2, the inner sum is4725, which is1250*8 -1875*4 +1275*2 -325=10,000 -7,500 +2,550 -325=10,000 -7,500=2,500 +2,550=5,050 -325=4,725.So, correct.But when I compute S=Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ] from1 to24, I get8,632,800.But when I compute manually up to k=2, I get4400, which is correct.Wait, but in my earlier calculation, I thought that S=1250A -1875B +1275C -325D, where A=Σ [ (-1)^k k³ ], etc.But when I compute for k=1 and k=2, the formula gives:For k=1: -2175For k=2: +4725Total:2,550But manually, it's-325 +4725=4400.Wait, so my mistake is in the way I separated the sums.Wait, because when I wrote S=1250A -1875B +1275C -325D, I assumed that A=Σ [ (-1)^k k³ ], but in reality, the expression is Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ].So, it's not simply1250A -1875B +1275C -325D, because A, B, C, D are sums of (-1)^k k³, etc., but the coefficients are different.Wait, no, actually, it is correct because:Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ]=1250 Σ [ (-1)^k k³ ] -1875 Σ [ (-1)^k k² ] +1275 Σ [ (-1)^k k ] -325 Σ [ (-1)^k ].So, S=1250A -1875B +1275C -325D.But when I computed for k=1 and k=2, I get:For k=1: -2175For k=2: +4725Total:2,550But manually, it's-325 +4725=4400.Wait, so discrepancy is because when I compute S=1250A -1875B +1275C -325D, I get2,550, but manually, it's4400.So, my approach is wrong.Wait, perhaps I made a mistake in the initial step.Wait, the inner sum is sum= (50k -25)*(50k² -50k +26)/2.But when I plug k=1, I get (25)*(26)/2=325, which is correct.Similarly, for k=2, (75)*(126)/2=4725, correct.So, the inner sum is correct.But when I expand it, I get1250k³ -1875k² +1275k -325.So, the expression is correct.But when I compute S=Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ] from1 to24, I get8,632,800.But when I compute manually up to k=2, I get4400, which is not matching with the formula's partial sum.Wait, perhaps I made a mistake in the way I separated the sums.Wait, let me think differently.Instead of separating into A, B, C, D, maybe I should compute the sum directly for each k and accumulate.But that would be tedious for k=1 to24.Alternatively, perhaps I can find a pattern or a generating function.Alternatively, perhaps I made a mistake in the initial assumption that the blocks are from (5(k-1))² +1 to (5k)^2.Wait, let me check the sign changes.The problem says the signs change every (5k)^2.So, the sign changes occur at25,100,225,...,14400.So, the first block is from1 to25, negative.Second block from26 to100, positive.Third block from101 to225, negative.And so on.So, each block is from (5(k-1))² +1 to (5k)^2, with sign (-1)^k.So, the number of terms in each block is (5k)^2 - (5(k-1))²=50k -25.So, the sum for each block is (-1)^k * sum from a to b, where a=(5(k-1))² +1, b=(5k)^2.Which is (-1)^k*(number of terms)*(a + b)/2.Which is (-1)^k*(50k -25)*(25(k-1)^2 +1 +25k²)/2.Which simplifies to (-1)^k*(50k -25)*(50k² -50k +26)/2.Which is (-1)^k*(2500k³ -3750k² +2550k -650)/2.Wait, but when I plug k=1, I get (-1)^1*(2500 -3750 +2550 -650)/2= (-1)*(2500 -3750= -1250 +2550=1300 -650=650)/2= (-1)*(650)/2= -325, which is correct.Similarly, for k=2: (-1)^2*(2500*8 -3750*4 +2550*2 -650)/2= (20,000 -15,000 +5,100 -650)/2= (20,000 -15,000=5,000 +5,100=10,100 -650=9,450)/2=4,725, which is correct.So, the formula is correct.But when I compute S=Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ] from1 to24, I get8,632,800.But when I compute manually up to k=2, I get4400, which is not matching with the formula's partial sum.Wait, because when I compute S=1250A -1875B +1275C -325D, I get8,632,800, but when I compute manually up to k=2, I get4400, which is not matching.Wait, perhaps I made a mistake in the way I separated the sums.Wait, no, the separation is correct because:Σ [ (-1)^k*(1250k³ -1875k² +1275k -325) ]=1250 Σ [ (-1)^k k³ ] -1875 Σ [ (-1)^k k² ] +1275 Σ [ (-1)^k k ] -325 Σ [ (-1)^k ].So, S=1250A -1875B +1275C -325D.But when I compute for k=1 and k=2, I get:A= (-1)^1*1³ + (-1)^2*2³= -1 +8=7.B= (-1)^1*1² + (-1)^2*2²= -1 +4=3.C= (-1)^1*1 + (-1)^2*2= -1 +2=1.D= (-1)^1 + (-1)^2= -1 +1=0.So, S=1250*7 -1875*3 +1275*1 -325*0=8,750 -5,625 +1,275=8,750 -5,625=3,125 +1,275=4,400.Which matches the manual sum.Wait, so when I compute A, B, C, D up to k=2, I get A=7, B=3, C=1, D=0.So, S=1250*7 -1875*3 +1275*1=8,750 -5,625 +1,275=4,400.Which is correct.But earlier, when I computed A=7344, B=300, C=12, D=0, I got S=8,632,800.But when I compute A, B, C, D up to k=24, I get A=7344, B=300, C=12, D=0.So, S=1250*7344 -1875*300 +1275*12=9,180,000 -562,500 +15,300=8,632,800.But when I compute manually up to k=2, I get4400, which is correct.So, the formula is correct.Therefore, the final answer is8,632,800.But let me think again. The series goes up to14400, which is(120)^2, so k=24.So, the sum is8,632,800.But let me check if this makes sense.The total number of terms is14400.The average value of the terms is roughly around7200.But the sum is8,632,800, which is roughly around7200*14400=103,680,000, which is way larger.Wait, no, that's not correct.Wait, the average value is sum divided by number of terms.So, sum=8,632,800, number of terms=14400.Average=8,632,800 /14,400≈600.Which seems reasonable because the terms go up to14400, but with alternating signs, so the average is around600.So, it seems plausible.Therefore, I think the final answer is8,632,800.

answer：First, I recognize that the farmer has 39 apples left after giving away 88 apples.To find out how many apples the farmer initially had, I need to add the number of apples he gave away to the number of apples he has left.So, I add 39 apples (left) to 88 apples (given away).39 + 88 equals 127.Therefore, the farmer initially had 127 apples.

answer：First, I need to determine the total number of boys in the camp, which I'll denote as ( T ).I know that 20% of the total boys are from School A. So, the number of boys from School A is ( 0.20 times T ).Out of these boys from School A, 30% study science. This means that 70% of the boys from School A do not study science.Given that there are 49 boys from School A who do not study science, I can set up the equation:[0.70 times (0.20 times T) = 49]Simplifying the equation:[0.14 times T = 49]To find ( T ), I divide both sides by 0.14:[T = frac{49}{0.14} = 350]Therefore, the total number of boys in the camp is 350.