Find the sum of the following arithmetic progressions.
${(x - y)}^{2}, (x^{2} + y^{2}), {(x + y)}^{2}, . . .$ to $n$ terms

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Solution

We know,

If $a =$ First term
$d =$ Common difference
$n =$ number of terms

then sum of n term
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

here
$a = (x - y)^{2}$

$d = (x^{2} + y^{2}) - (x - y)^{2} = x^{2} + y^{2} - x^{2} - y^{2} + 2 x y$

$d = 2 x y$

hence

sum $S_{n} = \frac{n}{2} [2 (x - y)^{2} + (n - 1) 2 x y]$

$= n ((x - y)^{2} + (n - 1) x y)$

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