Mathematical induction — Sum of r(2r+1)

Question

Prove by induction that for .

This practice question was inspired by Question 12(a) in the 2020 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 : LHS , RHS .
Finish the calculation, then check that the result meets the question’s conditions.Assume for ; add and factor to match the formula for .
State the final answer clearly in the form the question requested.Complete the induction.

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

Verification: Optional check: expand RHS and compare coefficients with .

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