Question

The first term of an infinitely decreasing GP is unity and its sum is S. Find the sum of the squares of the terms of the progression.

Answer 1

The first term of the series is $1$

Let the common ratio is $r$

So the series is $1,r,r^2,.....$

The sum is $S=\frac{1}{1-r}r=1-\frac{1}{S}$

Now the terms are squared so the series is $1,r^2,r^4,....$

The sum is given as $\frac{1}{1-r^2}\frac{1}{1+r}\frac{1}{1-r}\frac{1}{1+(1-\frac{1}{S})}S\frac{S}{2S-1}S\frac{S^2}{2S-1}$