For positive integer n, 4an=n^2+5n+6 and sn=∑ 1/ak k ∈ to (k=1, n). Then the value of 507(S2025) is

Question

For positive integer \(n, 4 a_n=n^2+5 n+6\) and

\(S_n=\sum_{k=1}^n \frac{1}{a_K}.\) Then the value of \(507\left(S_{2025)}\right.\) is

(1) 675

(2) 540

(3) 1350

(4) 725

Answer 1

Correct option is (1) 675  

\(S_n=\sum_{K=1}^n \frac{4}{K^2+5 K+6}\)

\(=\sum_{K=1}^n \frac{4}{(K+2)(K+3)}=4 \sum_{K=1}^n\left(\frac{1}{K+2}-\frac{1}{K+3}\right)\)

\(=4\left[\frac{1}{3}-\frac{1}{4}\right]\)

\(=4\left[\frac{1}{4}-\frac{1}{5}\right]\)

\(=4\left[\frac{1}{n+2}-\frac{1}{n+3}\right]\)

\(S_n=4\left[\frac{1}{3}-\frac{1}{n+3}\right]\)

\(S_{2025}=4\left[\frac{1}{3}-\frac{1}{2028}\right]\)

\(S_{2025}=4\left[\frac{675}{2028}\right]\)

\(507 \ S_{2025}=675\)