Mathematical induction — Partial fractions of the summand

Question

Decompose into partial fractions and hence explain the closed form .

This practice question was inspired by Question 12(c) in the 2021 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..
State the final answer clearly in the form the question requested.The sum telescopes to after cancellation, which rearranges to the given closed form.

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

Verification: Partial fractions motivate the inductive target.

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