Question

The ratio of sum of first three terms of a G.P to the sum of first six terms is $64:91$, the common ratio of G.P is

• A. $\frac{1}{4}$
• B. $\frac{3}{4}$
• C. $\frac{5}{4}$
• D. $\frac{7}{4}$

Answer 1

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

$\begin{array}{c}Weknow,\\ S_n=a\frac{r^n-1}{r-1}\\ So,\\ \frac{S_3}{S_6}=\frac{a\left(\frac{r^3-1}{r-1}\right)}{a\left(\frac{r^6-1}{r-1}\right)}=\frac{64}{91}\\ Assumingr\ne 1\\ \left(asr-1isindenominator\xi denomitorcannotbe0,r-1\ne 0,r\ne 1\right)\\ =\frac{a\left(\frac{r^3-1}{r-1}\right)}{a\left(\frac{r^6-1}{r-1}\right)}=\frac{64}{91}\\ =\frac{r^3-1}{r^6-1}=\frac{64}{91}\\ 91r^3-91=64r^6-64\\ 64r^6-91r^3+27=0\\ r=\frac{1}{4}\end{array}$