Mathematical induction — Related base

Question

Prove is divisible by 7 for by induction (or note , ).

This practice question was inspired by Question 12(f) in the 2022 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.Mod 7: , , so ; for odd get . Adjust statement to odd , or choose .
Finish the calculation, then check that the result meets the question’s conditions.Better clean variant: prove divisible by 7 for .
State the final answer clearly in the form the question requested.Base : ; step .

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

Verification: Inductive algebra producing an explicit factor of 7.

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