# Further Maths - Induction

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/induction/ · Updated: 2020-12-10 · Tags: further-maths, induction

See also:
- [Further Maths - Induction for Series](https://ollybritton.com/notes/a-level/further-maths/topics/induction-for-series/)
- [Further Maths - Induction for Divisibility](https://ollybritton.com/notes/a-level/further-maths/topics/induction-for-divisibility/)

##### What is induction??
A proof technique that shows a statement is true for natural numbers.

##### How could you use induction to prove you can climb as high as you like on a ladder??
* You can climb onto the bottom rung
* You can climb onto the next rung

##### What are the 4 steps for induction??
1. Basis
2. Assumption
3. Induction
4. Conclusion

##### What is the 'basis' step of induction??
Prove the statement for $n = 1$.

##### What is the 'assumption' step of induction??
Assume the statement is true for $n = k$.

##### What is the 'induction' step of induction??
Show that the general statement is true for $n = k+1$.

##### What is the 'conclusion' step of induction??
Summarising that the statement is true for all positive integers.

##### If using induction to prove something for even numbers, what could you do in the 'induction' step rather than $n = k+1$??
$$
n = k+2
$$

##### What are the 3 main topics of induction in the exam??
* Series
* Matricies
* Divisibility

##### What should you write after the induction step??
If the statement holds for $n = k$, it holds for $n = k + 1$.

##### What should you write for the conclusion step??
Since the statement holds for $n = 1$, the statement holds for all $n \in \mathbb{Z}^{++}$.

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