Question

Find the sum of the arithmetic progression:

(x – y)², (x² + y²), (x + y)², to 22 terms

Answer 1

In an A.P if the first term = a, common difference = d, and if there are n terms.

Then, sum of n terms is given by:

(x – y)²,(x² + y²), (x + y)²,… to n terms

First term (a) = (x – y)²

Common difference (d) = x² + y² – (x – y)²

= x² + y² – (x² + y² – 2xy)

= x² + y² – x² + y² + 2xy

= 2xy

Sum of n^th terms 

= n/2{2(x – y)² + (n – 1). 2xy}

= n{(x – y)² + (n – 1)xy}

∴ S_n = n{(x — y)² + (n — 1). xy)