## Related Geometric Series Problem

\[x\]

, 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

\[3x=1+x \rightarrow 2x=1 \rightarrow x = \frac{1}{2}\]

.