Proof By Induction

**parkerj3231** · September 29th 2009, 02:27 PM

Can somone solve this and show how to do it.

1/1x2 + 1/2x3 +...+ 1/n(n+1) = n/n+1

**Soroban** · September 29th 2009, 03:26 PM

Hello, parkerj3231!

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.

Thread: Proof By Induction

LinkBack

Thread Tools

Display

Proof By Induction

Similar Math Help Forum Discussions

Proof by induction?

Proof by Induction

Mathemtical Induction Proof (Stuck on induction)

Proof by induction

Proof with algebra, and proof by induction (problems)

Search Tags