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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Updated on: 10-Oct-2022

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