Question

If in an infinite G.P first term is equal to the twice of the sum of the remaining terms, then common ratio is

• A. $1$
• B. $2$
• C. $\frac{1}{3}$
• D. $-\frac{1}{3}$

Answer 1

The correct option is C

$\frac{1}{3}$

Explanation for correct answer:

Find the common ratio of the series:

Let $a,ar,a{r}^2\dots .\infty$ be the series.

Sum of the remaining terms in G.P $=ar+a{r}^2+\dots$

$S_{\infty }=\frac{ar}{1-r}\mathbf{[}\mathbf{\because }\mathbf{}{\mathbf{S}}_{\mathbf{\infty }}\mathbf{=}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}\mathbf{]}$

(Since, the first term is $ar$ in the above series)

Given that,

$\text{First term}=2\left(\text{sum of remaining terms}\right)$

$$\begin{gathered} a=\frac{2ar}{1-r} \\ \Rightarrow 1-r=2r \\ \Rightarrow 3r=1 \\ \Rightarrow r=\frac{1}{3} \end{gathered}$$
Hence, the correct option is C.