Induction proof of an equation issue

Question

Equation

Equation: 1/(1*2) + 1/(2*3) + 1/(2*4) + ... + 1/(n*(n+1)) = n/(n+1)

I need help with solving this equation. I cannot seem to be able to do anything with (n+1)/(n+2) that I get on the right side after induction step.

Answer 1

Hint $1$

$$\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$$

write every term like above and sum.

Hint $2$

If you really need use induction then use the standard approach for $n+1$:

$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{n(n+1)}+\frac{1}{(n+1)\cdot (n+2)}=\frac{n}{n+1}+\frac{1}{(n+1)\cdot (n+2)}$$

Finish the calculation and get

$$\frac{n+1}{n+2}$$