## Related Geometric Series Problem

A geometric series has first term
$x$
, common ratio
$x$
and sum
$S$
. A second geometric sequence has first term
$x^2$
, common ratio
$x^2$
and sum
$\frac{S}{3}$
. What is the value of
$x$
?
The sum of a geometric series
$S_{\infty}=\frac{a}{1-r}$
where
$a, \: r$
are the first term and the common ratio respectively.
For the first sequence,
$\frac{S}{3}=\frac{x}{1-x}$
.
For the second sequence,
$3S=\frac{x^2}{1-x^2}$
.
Divide the second of these expressions by the first.
$\frac{1}{3}=\frac{\frac{x^2}{1-x^2}}{\frac{x}{1-x}}=\frac{\frac{x^2}{(1+x)(1-x)}}{\frac{x}{1-x}}=\frac{x}{1+x}$

Hence
$3x=1+x \rightarrow 2x=1 \rightarrow x = \frac{1}{2}$
.