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Question
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
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Solution
(x − y)2, (x2 + y2), (x + y)2 ... to n terms
\[\text { We have }: \]
\[ a = {(x -y)}^2 , d = \left( x^2 + y^2 - {(x - y)}^2 \right) = 2xy\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ = \frac{n}{2}\left[ 2 {(x - y)}^2 + (n - 1)(2xy) \right]\]
\[ = \frac{n}{2} \times 2\left[ {(x -y)}^2 + (n - 1)(xy) \right]\]
\[ = n\left[ {(x - y)}^2 + (n - 1)(xy) \right]\]
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