# Proof by induction problem explanation

1. Feb 18, 2012

### xeon123

In this link (https://www.physicsforums.com/showthread.php?t=523874 [Broken]) there's an example of proof by induction.

Somewhere in the explanation there's the phrase:
Why they decided to multiply both sides by x+1?

Last edited by a moderator: May 5, 2017
2. Feb 18, 2012

### eumyang

It was assumed that
(1) $(1+x)^n\geq 1+nx$

And you want to prove that
(2) $(1+x)^{n+1}\geq 1+(n+1)x$

By multiplying both sides of (1) by (1+x), you'll get the LHS of the resulting inequality to match the LHS of (2).

$a^m a = a^m a^1 = a^{m+1}$ by the product property of exponents, so
$(1+x)^n (1+x) = (1+x)^{n+1}$.

3. Feb 18, 2012

### xeon123

So, the equation will be: $(1+x)^{n+1} \geq (1+x)(1+nx)$?

From the RHS I can't get 1+(n+1)x, or can I?

4. Feb 18, 2012

### HallsofIvy

Staff Emeritus
Well, what do you get when you multiply on the right?

5. Feb 18, 2012

### xeon123

$(1+x)(1+nx)$ is equal to $1+(n+1)x$?