2025 HSC Mathematics Extension 1
Question 12(c):
Mathematical induction
Prove for positive integers.
How to recognise this question
Question type: mathematical induction
- The command and notation point to mathematical induction.
- The requested response is a show that.
How to handle it: show the base case, state the assumption for n=k, and transform the k+1 case using that assumption.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Base case : both sides equal .
- Assume .
- For , add to the assumed sum.
- The result is , completing the induction.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Prove by induction that for .
Question 2
Prove by induction that divides for .
Question 3
Prove by induction that for every integer .