Mathematical induction — Related power induction

Question

Prove by induction that is divisible by for all integers .

This practice question was inspired by Question 12(d) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: show the base case, state the assumption for n=k, and transform the k+1 case using that assumption.Base : .
Finish the calculation, then check that the result meets the question’s conditions.Assume . Then .
State the final answer clearly in the form the question requested.Answer: true for all by induction.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Same base-and-step structure as the exam induction.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.