
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

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

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 $\displaystyle (1+x)^{n+1}\ge1+(n+1)x$.
If you have some problem, ask us.

I will skip the case for n=1.
Induction hypothesis and show true for P(k+1).
We have to show that $\displaystyle (x+1)^{k+1}\geq{1+(k+1)x}$
$\displaystyle (1+x)^{k}(1+x)\geq{(1+kx)(1+x)}$
$\displaystyle (1+x)^{k+1}\geq{1+x+kx+kx^{2}}$
$\displaystyle (1+x)^{k+1}\geq{1+(k+1)x+kx^{2}}$
The right side is clearly greater than 1+(k+1)x as we need.
P(k+1) is true and the induction is complete.