How to use Mathematical Induction to prove $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$?

Question

$$\frac{1}{1 \cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1}$$

What I have so far in the induction is:

$$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{k(k+1)} + \frac{1}{(k+1)(k+2)} = \frac{k}{k+1} + \frac{1}{(k+1)(k+2)}$$

But I'm not sure where to go from there . . .

Hint : Put the fraction on the same denominator. – Alain Remillard Dec 17 '15 at 20:57 — Alain Remillard, Dec 17 '15 at 20:57

score 4 · Accepted Answer · answered Dec 17 '15 at 20:58

4

Get a common denominator. Your last expression simplifies to $\dfrac{k^2 + 2k + 1}{(k+1)(k+2)}$. Now factor it.

answered Dec 17 '15 at 20:58

Umberto P.

score 3 · Answer 2 · 2015-12-19T10:27:31.777

3

$$\frac{k}{k+1}+\frac{1}{(k+1)(k+2)}=\frac{k(k+2)+1}{(k+1)(k+2)}=\frac{k+1}{k+2}$$

edited Dec 19 '15 at 10:27

answered Dec 17 '15 at 21:00

2 Answers2