Question

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

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

Answer 1

Answer: (D)

$\sum _{n=1}^{100}{a}_{2n+1}=200\Rightarrow a_3+a_5+a_7+....+a_{201}=200$
$\Rightarrow a{r}^2\frac{\left(r^{200}-1\right)}{\left(r^2-1\right)}=200$
$\sum _{n=1}^{100}{a}_{2n}=100\Rightarrow a_2+a_4+a_6+...+a_{200}=100$
$\Rightarrow \frac{ar\left(r^{200}-1\right)}{\left(r^2-1\right)}=100$
On dividing $r=2$
on adding $a_2+a_3+a_4+a_5+...+a_{200}+a_{201}=300$
$\Rightarrow r\left(a_1+a_2+a_3+....+a_{200}\right)=300$
$\Rightarrow \sum _{n=1}^{100}{a}_n=150$