Let GP is $a,ar,a{r}^2\dots$

$\sum _{n=1}^{100}{a}_{2n+1}=a_3+a_5+\dots a_{201}=200\Rightarrow \frac{a{r}^2(r^{200}-1)}{r^2-1}=200 \dots \left(1\right)$

$\sum _{n=1}^{100}{a}_{2n}=a_2+a_4+\dots a_{200}=100=\frac{ar\left(r^{200}-1\right)}{r^2-1}=100 \dots \left(2\right)$

Form $\left(1\right)$ and $\left(2\right)$, $r=2$

adding both,

$\Rightarrow a_2+a_3+\dots a_{200}+a_{201}=300\Rightarrow r\left(a_1+\dots a_{200}\right)=300$

$\sum _{n=1}^{200}\frac{300}{r}=150$