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

Question

For positive integers $n$, if $4a_n = n^2+5n+6$ and $S_n = \displaystyle\sum_{k=1}^{n}\frac{1}{a_k}$, then the value of $507\,S_{2025}$ is:

Accepted Answer

$$\frac{1}{a_k} = \frac{4}{k^2+5k+6} = \frac{4}{(k+2)(k+3)} = 4\left(\frac{1}{k+2}-\frac{1}{k+3}\right).$$

Telescoping: $S_n = 4\left(\dfrac13-\dfrac{1}{n+3}\right) = \dfrac{4n}{3(n+3)}$.

$$507\,S_{2025} = 507\times\frac{4\times2025}{3\times2028} = 675.$$

Solutions · 1