Problem
ALG-B1-M05-P009 Two-Step Telescoping
#9
★★★☆☆ Level 3 of 5
Find \(\sum_{k=1}^{n}\frac{1}{(k+1)(k+3)}\).
Decompose the fraction as \(\frac12\left(\frac1{k+1}-\frac1{k+3}\right)\).
\(\frac{1}{(k+1)(k+3)}=\frac12\left(\frac1{k+1}-\frac1{k+3}\right)\).
The sum is \(\frac12\left(\frac12+\frac13-\frac1{n+2}-\frac1{n+3}\right)\).
The cancellation is shifted by two steps.