Question

If the ratio of the sum of first three terms and the sum of first six terms of a$G.P.$ be $125:152$, then the common ratio $r$ is

• A. $\frac{3}{5}$
• B. $\frac{5}{3}$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A

$\frac{3}{5}$

Step 1. Find the common ratio $r$ is:

Sum of $n$ terms of $GP,S_n=\frac{a(r^n-1)}{(r-1)}$

$\Rightarrow$ $S_3=\frac{a(r^3-1)}{(r-1)}$

$\Rightarrow$ $S_6=\frac{a(r^6-1)}{(r-1)}$

Given $\frac{S_3}{S_6}=\frac{125}{152}$

$\Rightarrow$ $\frac{125}{152}$$=\frac{(r^3-1)}{(r^6-1)}$

$\Rightarrow$ $\frac{125}{152}$ $=\frac{(r^3-1)}{(r^3-1)(r^3+1)}$

$\Rightarrow$ $\frac{125}{152}$ $=\frac{1}{(r^3+1)}$

$\Rightarrow$ $125r^3+125=152$

$\Rightarrow$ $125r^3=27$

$\Rightarrow$ $r^3=\frac{27}{125}$

$\Rightarrow$ $r=\frac{3}{5}$

Hence, the option (A) is correct.