2025 ACJC Promo Q3

Given that \(\displaystyle\sum_{r=1}^{n}\dfrac{1}{r(r+1)}=\dfrac{n}{n+1}\).

1. Show that \(\displaystyle\sum_{r=2}^{n}\dfrac{1}{r(r-1)}=1-\dfrac{1}{n}\).[2]
2. Find \(\displaystyle\sum_{r=2}^{\infty}\left(\dfrac{1}{2^r}+\dfrac{1}{r(r-1)}\right)\).[3]