2022 YIJC Promo Q4

Express \(\frac{5r+8}{r\left( r+1 \right)\left( r+2 \right)}\) in partial fractions.[2]

Hence, find \(\sum\limits_{r=\,\,1}^{n}{\frac{5r+8}{r\left( r+1 \right)\left( r+2 \right)}}\), giving your answer in the form \(M-\frac{P}{n+1}-\frac{Q}{n+2}\), where \(M\), \(P\) and \(Q\) are constants to be determined.[3]

Use your answer in part (b) to find \(\sum\limits_{r=\,\,11}^{n+1}{\frac{5r+3}{r\left( {{r}^{2}}-1 \right)}}\).[3]