Question

Let $S_n$ represent sum to first $n$ terms of a G.P. whose first term is 1 and common ratio is 3 and $T_n$ represent the $n^{\text {th }}$ term of another GP. whose first term is 5 and common ratio is 5 . If $\displaystyle\sum_{n=1}^{\infty} \frac{S_{n+1}}{T_{n+2}}=\frac{p}{q}$ where $p$ and $q$ are in their lowest form $(p, q \in N)$, then find the value of $(p+q)$.

Answer 1

$S _{ n }=\frac{3^{ n }-1}{2} ; $
$ \therefore S _{ n +1}=\frac{3^{ n +1}-1}{2} $
$T _{ n }=5 \cdot 5^{ n -1} $
$\therefore T _{ n +2}=5^{ n +2}$
$\therefore \displaystyle\sum_{n=1}^{\infty} \frac{S_{n+1}}{T_{n+2}}=\displaystyle\sum_{n=1}^{\infty}\left(\frac{3 \cdot 3^n-1}{2 \cdot 5^n \cdot 25}\right)=\displaystyle\sum_{n=1}^{\infty} \frac{3}{50}\left(\frac{3}{5}\right)^n-\frac{1}{50} \displaystyle\sum_{n=1}^{\infty} \frac{1}{5^n}$
$=\frac{3}{50} \cdot \frac{\frac{3}{5}}{1-\frac{3}{5}}-\frac{1}{50} \cdot \frac{\frac{1}{5}}{1-\frac{1}{5}}=\frac{3}{50} \cdot \frac{3}{2}-\frac{1}{50} \cdot \frac{1}{4}=\frac{18}{200}-\frac{1}{200}=\frac{17}{200}$