IAL 2022 Oct Q2 | 國際數學站

(a) Write $\frac{3r+1}{r(r-1)(r+1)}$ in partial fractions.

(2)

(b) Hence find

\begin{align*} \sum_{r=2}^{n} \frac{3r+1}{r(r-1)(r+1)} \qquad n \geqslant 2\\[2mm] \end{align*}

giving your answer in the form

\begin{align*} \frac{an^2+bn+c}{2n(n+1)}\\[2mm] \end{align*}

where $a$ , $b$ and $c$ are integers to be determined.

(5)

\begin{align*} \sum_{r=15}^{20} \frac{3r+1}{r(r-1)(r+1)}\\[2mm] \end{align*}

(2)