For positive integers n, if 4a_n=(n^2+5n+6) and S_n= _k=1^n(1a_k), then the value of 507S_…

Question

For positive integers $n$, if $4{a}_{n}=({n}^{2}+5n+6)$ and ${S}_{n}=\sum _{k=1}^{n}(\frac{1}{{a}_{k}})$, then the value of $507{S}_{2025}$ is :

Accepted Answer

$\begin{gathered} {a}_{n} =\frac{{n}^{2}+5n+6}{4} \ {S}_{n} ={S}_{n}=\sum _{k=1}^{n}\frac{1}{{a}_{k}}=\sum _{1}^{n}\frac{4}{{k}^{2}+5k+6} \ =4\sum _{k=1}^{n}\frac{1}{(k+2)(k+3)} \ =4\sum _{k=1}^{n}\frac{1}{k+2}-\frac{1}{k+3} \ =4(\frac{1}{3}-\frac{1}{4})+4(\frac{1}{4}-\frac{1}{5})+\ldots \ldots .. \end{gathered}$

$\begin{gathered} 4(\frac{1}{n+2}-\frac{1}{n+3}) \ =4(\frac{1}{3}-\frac{1}{n+3}) \ =\frac{4n}{3(n+3)} \ {}^{507}{S}_{2025}=\frac{(507)(4)(2025)}{3(2028)} \ =675 \end{gathered}$

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