Answer
$$\sum_{i=1}^{n}\frac{i}{(i+1)(i+2)}$$
Work Step by Step
Given $$\frac{1}{2 \cdot 3}+\frac{2}{3 \cdot 4}+\dots+\frac{n}{(n+1)(n+2)}$$
The first term is $\dfrac{1}{2 \cdot 3}=\dfrac{1}{(1+1)(1+2)}$, the last term is $\dfrac{n}{(n+1)(n+2)}$, and we observe than the numerator is increasing by $1$, so
$$\frac{1}{2 \cdot 3}+\frac{2}{3 \cdot 4}+\dots+\frac{n}{(n+1)(n+2)}=\sum_{i=1}^{n}\frac{i}{(i+1)(i+2)}$$
