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The sum of infinite terms of a $G.P.$ is $x$ and on squaring the each term of it, the sum will be $y$, then the common ratio of this series is
$\frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}$
$\frac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}}$
$\frac{{{x^2} - y}}{{{x^2} + y}}$
$\frac{{{x^2} + y}}{{{x^2} - y}}$
Solution
(c) We have $\frac{a}{{1 – r}} = x$ …..$(i)$
and $\frac{{{a^2}}}{{1 – {r^2}}} = \frac{a}{{1 – r}}.\frac{a}{{1 + r}} = y$ …..$(ii)$
$ \Rightarrow $ $y = x.\frac{a}{{1 + r}} = x.\frac{{x(1 – r)}}{{1 + r}}$
$ \Rightarrow $$\frac{y}{{{x^2}}} = \frac{{1 – r}}{{1 + r}}$
$ \Rightarrow $ $\frac{{{x^2}}}{y} = \frac{{1 + r}}{{1 – r}}$
$ \Rightarrow $$\frac{{{x^2}}}{y}(1 – r) = 1 + r$
$\Rightarrow r[1+ \frac{{x^2}}{{y}}] = -1 + \frac{{x^2}}{{y}}$
$\Rightarrow r=\frac{{{x^2} + y}}{{{x^2} – y}}$
