# HSC Extension 1 and 2 Mathematics/3-Unit/HSC/Induction

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Induction is a form of proof useful for proving equations involving non-closed expressions (i.e., expressions with terms; sequences).

## Explanation[edit]

Induction involves first proving that the equation is true for , then proving true for (assuming for the purpose of the proof that the equation holds true for ). Since it is true for and true for , and also true for , it is true for . It follows that it is true for all positive integers .

### Examples[edit]

#### Proving the formula for the sum of a series[edit]

Q: Prove by mathematical induction that for all integers ,

A:

- When , , so it is true for
- Suppose that the statement is true for . That is, suppose that . This is sometimes called the
*induction hypothesis*. - Then prove the statement for (that is, prove that :
- It follows from parts 1 and 2 by mathematical induction that the statement is true for all positive integers .