Check or Evaluate this Summation

Question

How do I check or evaluate this summation$$\sum_{k=0}^n \frac{2(k+1)}{(n+1)(n+2)}=1$$ for $0\le k\le n$

Answer 1

$$\sum_{k=0}^n\frac{2(k+1)}{n+1)(n+2)}=\frac2{(n+1)(n+2)}\sum_{k=1}^{n+1}k=\frac2{(n+1)(n+2)}\cdot\frac{(n+1)(n+2)}2$$