Mathematical induction — Companion sum

Question

Prove by induction that for .

This practice question was inspired by Question 12(b) in the 2023 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 : both sides .
State the final answer clearly in the form the question requested.Assume for ; for : .

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

Verification: Classic induction template.

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