# Further Maths - Induction

## See also:

[[Further Maths - Induction for Series]]^{A}

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?

- Basis
- Assumption
- Induction
- 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 $n = k+1$?

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 $n \in \mathbb{Z}^{++}$.