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 $\sum _{n=1}^{\infty }\frac{S_{n+1}}{T_{n+2}}=\frac{p}{q}$ where $p$ and $q$ are in their lowest form $(p,q\in N)$, then find the value of $(p+q)$.

Answer 1

Answer: 217

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