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 \(r\ge 1\).

1. 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]
2. Hence find \(\sum\limits_{r=1}^{N}{\frac{5}{r\left( r+2 \right)}}\) in terms of \(N\).[4]
3. Given that \(\sum\limits_{r=N+1}^{\infty }{\frac{5}{r\left( r+2 \right)}}=\frac{11}{12}\), find the value of \(N\).[2]