Proof by induction $\frac1{1 \cdot 2} + \frac1{2 \cdot 3} + \frac1{3 \cdot 4} + \cdots + \frac1{n \cdot (n+1)} = \frac{n}{n+1}$ Need some help on following induction problem:

$$\dfrac1{1 \cdot 2} + \dfrac1{2 \cdot 3} + \dfrac1{3 \cdot 4} + \cdots + \dfrac1{n \cdot (n+1)} = \dfrac{n}{n+1}$$


Every question of the form: prove by induction that
$$\sum_{k=1}^n f(k)=g(n)$$
can be done by verifying two facts about the functions
$f$ and $g$:


*

*$f(1)=g(1)$
and

*$g(n+1)-g(n)=f(n+1)$.

