# Binomial theorem

The binomial theorem states that x,y and n,k, such as kn, we have:

## Inductive Proof

### Prerequisite information

• Have a knowledge of binomial coefficient.
• Have a knowledge of the principle of induction.
• Know that: x0=1.

### The proof

In order to show that the binomial theorem is true for n. We will first prove it is true for n=0, then for n by induction.

#### Proof for n=0

We have:

(x+y)0=1

and

We then deduce that the binomial theorem is true for n=0.

#### Inductive step

We know that for n=0, we have:

For n+1, we have:

This is why we have:

Because we know that the binomial theorem is true for n=0 and for n=n+1, we deduce by induction that it is true for n.