# Proof by Mathematical Induction

# Proof by mathematical induction^{[1]}[edit | edit source]

Mathematical induction is the process of verifying or proving a mathematical statement is true for all values of within given parameters. For example:

We are asked to prove that is divisible by 4. We can test if it's true by giving values.

So, the first 5 values of n are divisible by 4, but what about all cases? That's where mathematical induction comes in.

Mathematical induction is a rigorous process, as such all proofs must have the same general format:

- Proposition – What are you trying to prove?
- Base case – Is it true for the first case? This means is it true for the first possible value of .
- Assumption – We assume what we are trying to prove is true for a general number. such as , also known as the induction hypothesis.
- Induction – Show that if our assumption is true for the ( term, then it must also be true for the ( term.
- Conclusion – Formalise your proof.

There will be four types of mathematical induction that you will come across in FP1:

- Summation of series;
- Divisibility;
- Recurrence relations;
- Matrices.

## Example of a proof by summation of series[edit | edit source]

(Empty)

## Example of a proof by divisibility[edit | edit source]

Proposition:

Notice our parameter, . This means that what we want to prove must be true for all values of which belong to the set (denoted by ) of positive integers ().

Base case:

Assumption (Induction Hypothesis): Now we let where is a general positive integer, and we assume that .

Remember that .

Induction: Now we want to prove that the term is also divisible by 4

Hence:

This is where our assumption comes in, if then 4 must also divide .

So:

Now we've shown and thus . This implies that because you have successfully shown that 4 divides , where is a general, positive integer () and also the consecutive term after the general term ()

Conclusion:

## Example of a proof by recurrence relations[edit | edit source]

(Empty)

## Example of a proof by matrices[edit | edit source]

(Empty)