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← 2020 HSC Extension 1 paper Question 12(a)

Variation 2 · answer guide

Mathematical induction — Cube sum

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.

Step by step

  1. 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 .
  2. State the final answer clearly in the form the question requested.Inductive step: add and factor .

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

Verification: Standard inductive algebra for the square of the triangular numbers.

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

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