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

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For positive integer \(n,4{a}_{n}={n}^{2}+5n+6\) and \({S}_{n}=\sum _{k=1}^{n}\left(\frac{1}{{a}_{k}}\right)\), then the…