Mathematical induction — A sum identity

Question

Prove by induction that for .

This practice question was inspired by Question 12(c) in the 2025 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 case : both sides equal .
Simplify the previous line carefully, keeping exact values where possible.Assume the result for .
Simplify the previous line carefully, keeping exact values where possible.Add : .
Finish the calculation, then check that the result meets the question’s conditions.This is the required formula for .
State the final answer clearly in the form the question requested.Final answer: The identity holds for every positive integer.

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

Verification: The base case and implication from k to k+1 complete the induction.

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