Consider three infinite geometric progressions \( x=a-a r+a r^{2}-a r^{3}+\ldots . . \infty \) \...
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0Consider three infinite geometric progressions \( x=a-a r+a r^{2}-a r^{3}+\ldots . . \infty \) \( y=a+a r^{2}+a r^{4}+\ldots \ldots \infty \) and \( z=a+a r^{3}+a r^{6}+\ldots \ldots \infty \) where \( |r|1 \).
Then prove
\[
\frac{y}{z}=\frac{r^{2}+r+1}{r+1}
\]
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