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Q.
The first term of an infinitely decreasing GP. is unity and its sum is S. The sum of the squares of the terms of the progression is :
Sequences and Series
Solution:
$ S = a + ar + ar ^2+$ ........
$S=1+r+r^2+r^3+\ldots \ldots . . . . . . . (a=1)$
$S =\frac{1}{1- r } \Rightarrow 1- r =\frac{1}{ S } \Rightarrow r =1-\frac{1}{ S } \Rightarrow \frac{ S -1}{ S }= r$
$\text { Now } E = a ^2+( ar )^2+\left(a ar ^2\right)^2+ $......
$E = a ^2+ a ^2 r ^2+ a ^2 r ^4+ $ .....
$=\frac{ a ^2}{1- r ^2}=\frac{1}{1-\left(\frac{ S -1}{ S }\right)^2}=\frac{ S ^2}{ S ^2- S ^2-1+2 S }=\frac{ S ^2}{2 S -1} $