Find the sum of n term of the A P

To Find: The sum of n terms of the given AP.

Sum of n terms of an AP with first term a and common difference d is given by

$S=\frac{n}{2}[2 a+(n-1) d]$

Here $a=x-y, d=2 x-y$

$\Rightarrow S=\frac{1}{x+y} \times \frac{n}{2} \times[2 x-2 y+(n-1)(2 x-y)]$

$\Rightarrow S=\frac{n}{2(x+y)}[2 x-2 y+n(2 x-y)-2 x+y]$

$\Rightarrow S=\frac{n}{2(x+y)}[n(2 x-y)-y]$

The sum of the series is $\frac{\mathrm{n}}{2(\mathrm{x}+\mathrm{y})}[\mathrm{n}(2 \mathrm{x}-\mathrm{y})-\mathrm{y}]$