2022 NYJC Promo Q6
A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$,… is such that ${{u}_{r}}=\frac{1}{r}-\frac{1}{r+2}$ where $r\ge 1$.
(i)
Show that ${{u}_{r}}=\frac{A}{r\left( r+2 \right)}$ for a constant $A$ to be determined. Describe the behaviour of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$,… .
[2]
(ii)
Hence find $\sum\limits_{r=1}^{N}{\frac{5}{r\left( r+2 \right)}}$ in terms of $N$.
[4]
(iii)
Given that $\sum\limits_{r=N+1}^{\infty }{\frac{5}{r\left( r+2 \right)}}=\frac{11}{12}$, find the value of $N$.
[2]
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