Question

Let $a_n$ be the $n^{th}$ term of a G. P. of positive terms. If $\sum _{n=1}^{100}{a}_{2n+1}=200$ and $\sum _{n=1}^{100}{a}_{2n}=100$, then $\sum _{n=1}^{200}{a}_n$ is equal to :

• A. $300$
• B. $175$
• C. $225$
• D. $150$

Answer 1

The correct option is D $150$

Let $a,ar,a{r}^2,\cdots$ be the G.P.
$\sum _{n=1}^{100}{a}_{2n+1}=a_3+a_5+\cdots +a_{201}$
$\Rightarrow 200=a{r}^2+a{r}^4+\cdots +a{r}^{200}\Rightarrow 200=\frac{a{r}^2(r^{200}-1)}{r^2-1}\cdots \left(1\right)$

Also, $\sum _{n=1}^{100}{a}_{2n}=100$
$\Rightarrow 100=a_2+a_4+\cdots +a_{200}\Rightarrow 100=ar+a{r}^3+\cdots +a{r}^{199}$

$\Rightarrow 100=\frac{ar(r^{200}-1)}{r^2-1}\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, $r=2$
And $\sum _{n=1}^{100}{a}_{2n+1}+\sum _{n=1}^{100}{a}_{2n}=300$
$\Rightarrow a_2+a_3+a_4\cdots +a_{200}+a_{201}=300$
$\Rightarrow ar+a{r}^2+a{r}^3+\cdots +a{r}^{200}=300\Rightarrow r(a+ar+a{r}^2+\cdots +a{r}^{199})=300$
$\Rightarrow 2(a_1+a_2+a_3+\cdots +a_{200})=300$
$\therefore \sum _{n=1}^{200}{a}_n=150$