HSC Extension 1 and 2 Mathematics/3-Unit/HSC/Induction
Induction is a form of proof useful for proving equations involving non-closed expressions (i.e., expressions with terms; sequences).
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 .
Proving the formula for the sum of a series
Q: Prove by mathematical induction that for all integers ,
- 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 .