As you get closer towards university level maths, **proof** becomes very important.

**Inductive proof** can be used if we suspect something to be true. Metaphorically it is like proving an infinite ladder exists by first showing that the bottom rung of the ladder exists (the **initial step**) and then showing that if any rung of the ladder exists then the rung above it must necessarily also exist (the **inductive step**). If we have shown these two steps to be true, then we have proved that the ladder exists.

This process is easiest seen by way of some examples

**Worked Examples – Sum of Series**

- Prove, by mathematical induction, that
- for all n≥1

This method of proof can be used in many different areas of mathematics, and not just summation of series.

**Worked Examples – Differentiation**

- Prove, by mathematical induction, that
- If y = e
^{2x}, then - The 2nth derivative of y = cos(1-2x) is

- If y = e

**Worked Examples – Recurrence Relatio**ns

- Prove the following two problems using mathematical induction:
- If , where u
_{1}= 5, show that u_{n}>2 for all n≥1 - If , where u
_{1}= 1, show that u_{n}= 2 x 3^{n-1}-1 for all n ≥ 1

- If , where u

**Worked Example – Matrices**

Prove by induction that

**Exercise**

**Answers**

**Proof of divisibility**

Consider proving that f(n) = 2^{2n+2}+5 is divisible by 3 for all n≥0. For our inductive step we consider f(k+1) – f(k)

- f(k+1) – f(k) = 2
^{2(k+1)+2}+5 – 2^{2k+2}+5 = 2^{2k+4}– 2^{2k+2} - This equal 2
^{2k}(2^{4}-2^{2}) = 12 x 2^{2k}, which is clearly divisible by 3.

So as f(k+1) – f(k) is divisible by 3, if we assume f(k) is divisible by 3, it follows that f(k+1) is also divisible by 3. So we can complete a full inductive proof.

**Worked Example**

A function is defined as f(n) = 3^{n+2} + 5. Using mathematical induction, prove that 3^{n+2}+5 is always divisible by 2 for n≥0.

The exact same method can be used with polynomials

**Worked Example**

A function is given as f(n) = (n+2)^{3} + (2n+1)^{3}. Prove by induction that f(n) is divisible by 3 for all values n ≥ 0.

**Exercise & General Exercises**