Question

If each term of an infinite G.P. is thrice the sum of the terms following it, then the common ratio of the G.P. is .

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{4}$

Let a be the first term and r be the common ratio of the G.P.
$a_n=3\left(a_{n+1}+a_{n+2}+a_{n+3}+.....\infty \right]\Rightarrow {\mathrm{ar}}^{n-1}=3\left({\mathrm{ar}}^n+{\mathrm{ar}}^{n+1}+{\mathrm{ar}}^{n+2}+.....\infty \right]\Rightarrow {\mathrm{ar}}^{n-1}=3{\mathrm{ar}}^n\left(1+r+r^2+.....\infty \right]\Rightarrow \frac{{\mathrm{ar}}^{n-1}}{{\mathrm{ar}}^n}=3\times \frac{1}{1-r}(\because (r|<1)\Rightarrow \frac{1}{r}=\frac{3}{1-r}\Rightarrow 1-r=3r\Rightarrow 4r=1\Rightarrow r=\frac{1}{4}$