Question

In a $G.P$. the ratio of the sum of the first eleven terms to the sum of last eleven terms is $\frac{1}{8}$ and the ratio of the sum of all terms without the first nine to the sum of all the terms without the last nine is $2$. Then the number of terms of the $G.P$ is

• A. $15$
• B. $43$
• C. $38$
• D. $56$

Answer 1

The correct option is C $38$

Let the terms of the GP be $a,ar,a{r}^2....a{r}^{n-1}$
By the given conditions, we have
$\frac{a+ar+...a{r}^{10}}{a{r}^{n-11}+a{r}^{n-10}...a{r}^{n-1}}=\frac{1}{8}$
$⟹\frac{1}{r^{n-11}}=\frac{1}{8}$
and , $\frac{a{r}^9+a{r}^{10}+...a{r}^{n-1}}{a+ar...a{r}^{n-10}}=2$
$⟹r^9=2⟹r^{27}=8$
for $r^{n-11}=8$ , $n-11=27$
Hence, $n=38$