- A-Level Further Maths

# Proof by Induction

**Proof by induction** is used to prove that a general statement is true for all positive integer values. All proofs by **mathematical induction** follow four basic steps:

**Prove**that the general statement is true when*n*= 1**Assume**the general statement is true for*n*= k**Show**that, if it is true for*n*= 1, the general statement is also true for*n*= k+1**Conclude**that the general statement is true whenever*n*∈**ℕ**

All four steps must be shown clearly in your workings: prove, assume, show, conclude.

## Proving Sums

Often, questions will use the standard results for the sums of r, r² and r³. Regardless, the method for all sums is the same and follows the four steps above.

### Quick Tip

For the 3rd step, it is generally best to **write the last line out first** using the general function - just substitute (k+1) into it. You know that this is the answer you want to reach, so use it as a target to help you.

## Proving Divisibility Results

Again, follow the four standard steps for proof by induction. For divisibility results, make step 2 equal any general multiple of the divisor:

## Proof Using Matrices

Exactly the same four steps apply: