2022 ASRJC P1 Q5
(i)
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]
(ii)
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]
(iii)
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]
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