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Find the sum of the following arithmetic progressions:$ (x-y)^{2},\left(x^{2}+y^{2}\right),(x+y)^{2}, \ldots, $ to $ n $ terms
Given:
Given A.P. is \( (x-y)^{2},\left(x^{2}+y^{2}\right),(x+y)^{2}, \ldots, \)
To do:
We have to find the sum of the given A.P. to $n$ terms.
Solution:
Here,
\( a=(x-y)^{2}, d=x^{2}+y^{2}-(x-y)^{2}=x^{2}+ y^{2}-x^{2}-y^{2}+2 x y \)
\( \Rightarrow d=2 x y \)
${S}_{n}=\frac{n}{2}[2 a+(n-1) d]$
$=\frac{n}{2}\left[2(x-y)^{2}+(n-1)(-2 x y)\right]$
$=\frac{n}{2}\left[2(x-y)^{2}-2(n-1) x y\right]$
$=\frac{n}{2} \times 2\left[(x-y)^{2}-(x-1) x y\right]$
$=n\left[(x-y)^{2}-(x-1) x y\right]$
The sum of the given A.P. to $n$ terms is $n[(x-y)^2-(x-1)xy]$.
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