2025 RVHS P1 Q4

It is given that \(\sum_{r=1}^{n} \frac{1}{r(r+1)} = 1 - \frac{1}{n+1}\).

1. Find \(\sum_{r=1}^{N} \frac{1}{(r+1)(r+2)}\) in terms of \(N\).[3]
2. It is given that \[8 \sum_{r=k+1}^{\infty} \frac{1}{r(r+1)} = \sum_{r=1}^{k} \frac{1}{r(r+1)}.\] Find the value of \(k\).[3]