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

$\begin{array}{l}{a}_3+a_5\dots a_{201}=200 a_2+a_4\dots a_{200}=100\\ a{r}^2+a{r}^4\dots a{r}^{200}=200,ar+a{r}^3\dots a{r}^{199}=100\\ \frac{a{r}^2\left(r^{200}-1\right)}{r^2-1}=200 \left(1\right)\frac{ar\left(r^{200}-1\right)}{r^2-1}=100\\ \left(1\right)÷\left(2\right)\\ \frac{{\displaystyle \frac{a{r}^2\left(r^{200}-1\right)}{r^2-1}}}{\frac{ar\left(r^{200}-1\right)}{r^2-1}}=2\\ l\left(1\right)+\left(2\right)\\ a_2+a_3+a_4\dots .+a_{201}=300\\ ar+a{r}^2+a{r}^3\dots .a{r}^{200}=300\\ r\left(a+ar+a{r}^2\dots \dots a{r}^{199}\right)=300\\ r\left(a_1+a_2+a_3\dots a_{200}\right)=300\\ \sum _{r=1}^{200}{a}_r=\frac{300}{r}=\frac{300}{2}=150\end{array}$