What does the term 'inductive hypothesis' mean in an induction proof?

Conditions

• Used within the proof of the inductive step.
• Assumes $P(k)$ is true for a fixed arbitrary k.
• Required to prove $P(k+1)$.

Reasoning, step by step

1. State the goal: prove $P(k)$ implies $P(k+1)$.
2. Assume $P(k)$ is true; this assumption is the inductive hypothesis.
3. Use this assumption to manipulate the expression for $P(k+1)$.
4. Derive that $P(k+1)$ is true based on the hypothesis.
5. Conclude the inductive step.

Example

The narrator says, 'The assumption $P(k)$ is labeled the inductive hypothesis. This term identifies the temporary premise used inside the inductive step, not the final theorem itself.' In the example, $P(k)$: $2 + 4$ + ... + $2k = k(k+1)$ is assumed to prove the next case.

Common misconceptions

• Thinking that assuming $P(k)$ already proves the theorem for all n.
• Confusing the inductive hypothesis with the base case.

Watch the explanation

Explore next

• How is the inductive step executed for the sum of even integers?
• Why is the inductive hypothesis necessary?