Algebraic Proof Using Induction Let $n$ be a positive integer, and let $x \ge -1$. Prove, using induction, that
$$(1 + x)^n \ge 1 + nx.$$
I don't know what to do, I can't expand the left side.  I'm not familiar with induction, so can someone please provide an answer?  Thanks!
 A: Hint: $(1+x)^{n+1}=(1+x)^n\cdot(1+x)\geq(1+nx)(1+x)=1+(n+1)x+nx^2$
A: I know you asked for an inductive proof.  Nonetheless, here is a non-inductive proof, using AM-GM.  Suppose that $1+nx\geq 0$ (the case $1+nx<0$ being trivial).  Then, by AM-GM,
$$1+x=\frac{(n-1)\cdot 1+1\cdot (1+nx)}{(n-1)+1}\geq \left(1^{n-1}(1+nx)^1\right)^{\frac{1}{(n-1)+1}}=(1+nx)^{\frac{1}{n}}\,.$$
The result follows immediately (and the equality case is easily seen to be $x=0$ and $n=1$).  Note that $n$ can be any real number greater than or equal to $1$.
A: Induction always has two steps.
1) Basic step.  Show it is true for $n = 1$.  (Although sometime for $n=0$ or another number--- but its always a base case.  As we want to prove it for all positive integers we want to start by proving it is true for $n = $ the smallest positive integer).
If $n = 1$ then
$(1 + x)^n = (1 + x)^1 = 1 + x = 1 + 1*x = 1 + nx \ge 1 + nx$.
So it is true for $n = 1$.
2) Inductive step.  Show that if is is true for some $n = k$.  Then it is true also for $n = k + 1$.
[If we can show this then we know since it is true for $n=k =1$ it will also have to be true for $n = 2; n= 3; n =4;.... n = j; n = j+1; n = j+ 2......$ and therefore be true for all positive integers.]
So if $(1 + x)^k \ge 1 + kx$ then...
$(1 + x)^{k+1} = (1+x)(1+x)^k$.  
Now we know $(1 + x)^k \ge 1 + kx$
And as $x \ge -1$ we know $1 + x \ge 0$
So $(1+x)(1+x)^k \ge (1+x)(1 + kx)$
$(1+x)(1+kx) = 1 + x + kx + kx^2$
$ = 1 + (k+1)x + kx^2 \ge 1 + (k+1)x$.
So $(1+x)^{k+1} \ge 1+ (k+1)x$
So we have proven that if it is true for $n = k$ it is true for $n = k + 1$.
....
Therefore we are done.  (Because it is true for $n = 0$ so it is true for $n =1$.  It is true for $n = 1$ so it is true for $n = 2$.  It is true for $n = 2$ so it is true for $n = 3$.  It is true for.....)
