# Proof By Induction

• Sep 29th 2009, 02:27 PM
parkerj3231
Proof By Induction
Can somone solve this and show how to do it.

1/1x2 + 1/2x3 +...+ 1/n(n+1) = n/n+1
• Sep 29th 2009, 03:26 PM
Soroban
Hello, parkerj3231!

Quote:

Prove by induction: . $\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \hdots + \frac{1}{n(n+1)} \;=\;\frac{n}{n+1}$

Verify $S(1)\!:\;\;\frac{1}{1\cdot2} \:=\:\frac{1}{1+1}$ . . . true.

$\text{Assume }S(k)\!:\;\;\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \hdots + \frac{1}{k(k+1)} \;\;=\;\;\frac{k}{k+1}$

Add $\frac{1}{(k+1)(k+2)}$ to both sides:

. . $\underbrace{\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \hdots + \frac{1}{(k+1)(k+2)}}_{\text{This is the left side of }S(k+1)} \;\;=\;\;\frac{k}{k+1} + \frac{1}{(k+1)(k+2)}$

The right side is: . $\frac{k}{k+1}\cdot{\color{blue}\frac{k+2}{k+2}} + \frac{1}{(k+1)(k+2)} \;\;=\;\;\frac{k(k+2) + 1}{(k+1)(k+2)} \;\;=\;\;\frac{k^2 + 2k + 1}{(k+1)(k+2)}$

. . . . . . . . . . $= \;\;\frac{(k+1)^2}{(k+1)(k+2)} \;\;=\;\;\frac{k+1}{k+2} \quad\Leftarrow\;\text{ This is the right side of }S(k+1)$

We have proved $S(k+1)$ . . . The inductive proof is complete.