If sum of all terms of an infinite GP is times the sum of odd terms. Then, common ratio is

JEE MainMathematicsModerateSequences and Series

If sum of all terms of an infinite GP is $\frac{1}{5}$ times the
sum of odd terms. Then, common ratio is

Show answer and solution

Let S denotes the sum of all terms and S_1 denotes the sum of
odd terms.

S_{1} = \frac{a}{1 - r^{2}}

(i.e. sum of odd terms)
Given,

S = \frac{1}{5}S_{1} \Rightarrow \frac{a}{1 - r} = \frac{1}{5} \times \frac{a}{1 - r^{2}}

\Rightarrow 1 + r = \frac{1}{5} \Rightarrow r = - \frac{4}{5}

Source: Grass-G / M-23 Mathematics Paper 31 Dec FST.docx

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