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