IAL 2021 June Q1

(a) Express

\begin{align*} \frac{2}{r(r^2 - 1)} \end{align*}

in partial fractions.

(3)

(b) Hence find, in terms of $n$ ,

\begin{align*} \sum_{r=2}^{n} \frac{1}{r(r^2 - 1)} \end{align*}

Give your answer in the form

\begin{align*} \frac{n^2 + An + B}{Cn(n + 1)} \end{align*}

where $A$ , $B$ and $C$ are constants to be found.

(5)