Question

Let common ratio of a G.P. be $\cot \theta$ and sum of its infinite number of terms is $10$. A new G.P. is formed by taking cube of each of the terms of given series. If sum of infinite terms of new series is $\frac{1000}{7}$, then the number of possible value(s) of $\theta \in [0,5\pi ]$ is

Answer 1

Let $a$ be the first term of the series.
Then $10=\frac{a}{(\cot \theta -1)}(|\cot \theta |<1)$
$\Rightarrow a=10(1-\cot \theta )$
Hence, series is $10(1-\cot \theta ),10\cot \theta (1-\cot \theta )\dots \dots \infty$
New series will be ${10}^3{(1-\cot \theta )}^3,{10}^3{\cot }^3\theta {(1-\cot \theta )}^3\dots \dots \infty$
Sum of new series $=\frac{{10}^3{(1-\cot \theta )}^3}{1-{\cot }^3\theta }$
$\Rightarrow \frac{{10}^3{(1-\cot \theta )}^3}{1-{\cot }^3\theta }=\frac{1000}{7}$
$\Rightarrow 7-21\cot \theta +21{\cot }^2\theta -7{\cot }^3\theta =1-co{t}^3\theta$
$\Rightarrow 2{\cot }^3\theta -7{\cot }^2\theta +7\cot \theta -2=0$
$\Rightarrow \cot \theta =1,2,\frac{1}{2}$
But $|\cot \theta |<1$
$\Rightarrow \cot \theta =\frac{1}{2}$


Hence, number of solutions in $[0,5\pi ]$ is $5.$