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