Mathematics - Sequence & Series Question with Solution | TestHub

$\begin{array}{rl}& S_n=\sum _{k=1}^n\frac{4}{K^2+5k+6}\\ & =\sum _{k=1}^n\frac{4}{(K+2)(K+3)}=4\sum _{K=1}^n\left(\frac{1}{K+2}-\frac{1}{K+3}\right)\end{array}$
$\begin{array}{rl}& =4\left[\frac{1}{3}-\frac{1}{4}\right]\\ & =4\left[\frac{1}{4}-\frac{1}{5}\right]\\ & =4\left[\frac{1}{n+2}-\frac{1}{n+3}\right]\\ & S_n=4\left[\frac{1}{3}-\frac{1}{n+3}\right]\end{array}$
$\begin{array}{rl}& S_{2025}=4\left[\frac{1}{3}-\frac{1}{2028}\right]\\ & S_{2025}=4\left[\frac{675}{2028}\right]\\ & 507S_{2025}=675\end{array}$