Question

If the ratio of the sum of first three terms and the sum of first six terms of a G.P. be 64 : 189, then the common ratio r is _______.

• A. $\frac{5}{4}$
• B. $\frac{5}{3}$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A

$\frac{5}{4}$

Let the first term be 'a'.

Second Term would be 'ar'.

Third term would be $a{r}^2.$

$T_n=a{r}^{n-1}$

Given,

$\frac{a+ar+a{r}^2}{a+ar+a{r}^2+a{r}^3+a{r}^4+a{r}^5}=\frac{64}{189}$

$\frac{a(1+r+r^2)}{a(1+r+r^2)+a(r^3+r^4+r^5)}=\frac{64}{189}$

$\frac{1+r+r^2}{1+r+r^2+r^3+r^4+r^5}=\frac{64}{189}$

$\Rightarrow \frac{1+r+r^2}{(1+r+r^2)+r^3(1+r+r^2)}=\frac{64}{189}$

$\Rightarrow \frac{1+r+r^2}{(1+r+r^2)(r^3+1)}=\frac{64}{189}$

$\Rightarrow \frac{1}{1+r^3}=\frac{64}{189}\Rightarrow r^3=\frac{125}{64}$
$\therefore r=\frac{5}{4}$