# induction

• Apr 17th 2008, 04:37 PM
daaavo
induction
hi there,

I'm struggling with the last bit of this induction question...

Q: Let x>-1. Prove by induction that:

(1+x)^n > 1+nx

for every integer n>1
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So far I've got:

Let P(n) be the statement (1+x)^n > 1+nx

(1+x)^1 > 1+nx = 1+x

Therefore P(1) is true.

Assume true for some n+1

I cannot figure out where to go from here..... could you possibly help me?

Cheers
• Apr 17th 2008, 05:21 PM
arbolis
tip
Quote:

Assume true for some n+1
If I'm not wrong, it should be "Assume that p(n) is true, let's see if p(n)⇒p(n+1). "
So you must start assuming that p(n) is right and you will be in need to use the p(n) formula. If it implies p(n+1), then p(n+1) is right, then p(n) is right for all n >1.
In other words, using the formula (1+x)^n > 1+nx, you must reach $(1+x)^{n+1}\ge1+(n+1)x$.
If you have some problem, ask us.
• Apr 17th 2008, 06:05 PM
galactus
I will skip the case for n=1.

Induction hypothesis and show true for P(k+1).

We have to show that $(x+1)^{k+1}\geq{1+(k+1)x}$

$(1+x)^{k}(1+x)\geq{(1+kx)(1+x)}$

$(1+x)^{k+1}\geq{1+x+kx+kx^{2}}$

$(1+x)^{k+1}\geq{1+(k+1)x+kx^{2}}$

The right side is clearly greater than 1+(k+1)x as we need.

P(k+1) is true and the induction is complete.