2022 ASRJC P1 Q5

By considering \({{u}_{n}}-{{u}_{n+1}}\), where \({{u}_{n}}=\frac{1}{n\left( n+1 \right)\left( n+2 \right)}\), find \(\sum\limits_{n=1}^{N}{\frac{1}{n\left( n+1 \right)\left( n+2 \right)\left( n+3 \right)}}\) in terms of \(N\).[3]

Hence or otherwise, find \(\sum\limits_{n=5}^{N+3}{\frac{1}{n\left( n-1 \right)\left( n-2 \right)\left( n-3 \right)}}\).[3]

Deduce that \(\frac{1}{{{6}^{2}}}+\frac{1}{{{12}^{2}}}+\frac{1}{{{20}^{2}}}+\frac{1}{{{30}^{2}}}+\frac{1}{{{42}^{2}}}+...\) is less than \(\frac{1}{18}\). Show your workings clearly.[3]