1. ## Induction Inequality

Prove by induction
(1+h)^n≥ 1 + nh + ((n(n-1))/2)h^2

tried something, hope it helps

The inequality is true for $\displaystyle n=0$ (easy to show)

Assume the inequality holds for $\displaystyle n$

$\displaystyle (1+n)^{n+1}$
$\displaystyle =(1+h)(1+h)^{n}$
$\displaystyle \ge (1+h)(1+nh+\frac{n(n-1)h^{2}}{2})$
$\displaystyle =1+nh+\frac{n(n-1)h^{2}}{2}+h+nh^{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle =1+(n+1)h+\frac{n(n+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle \ge 1+(n+1)h+\frac{n(n+1)h^{2}}{2}$ if $\displaystyle h$ is positive

3. I'm a little confused as to how you went from the third line from the bottom to the next line.

4. $\displaystyle =1+nh+\frac{n(n-1)h^{2}}{2}+h+nh^{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle =1+nh+h+\frac{(n^{2}-n)h^{2}}{2}+\frac{2nh^{2}}{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle =1+(n+1)h+\frac{(n^{2}-n+2n)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle =1+(n+1)h+\frac{(n^{2}+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle =1+(n+1)h+\frac{n(n+1)h^{2}}{2}+\frac{n(n-1)h^{3}}{2}$
$\displaystyle \ge 1+(n+1)h+\frac{n(n+1)h^{2}}{2}$ if $\displaystyle h$ is positive

is this ok now?