2023 HSC Mathematics Extension 1
Question 12(b):
Mathematical induction
Use mathematical induction to prove that
for all 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 : LHS ; RHS . Equal, so true for .
- Inductive hypothesis: assume true for :
- .
- Inductive step for :
- LHS .
- RHS for : .
- LHS RHS, so true for . By induction, true for all integers .
- NESA: 3 marks complete proof; 2 for establishing the inductive step; 1 for the base case.
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.
Question 3
Verify the 2023 identity numerically for and .