2024 HSC Mathematics Extension 1
Question 12(d):
Mathematical induction
Use mathematical induction to prove that is divisible by 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 : , which is divisible by . True for .
- Inductive hypothesis: assume true for , i.e. for some integer . Equivalently .
- Inductive step for : .
- Since is an integer, is divisible by .
- By mathematical induction the statement holds for all integers .
- NESA 12(d): 3 marks for a correct proof; 2 marks for establishing the inductive step; 1 mark 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
Use mathematical induction to prove that is divisible by for all integers .
Question 2
Prove by induction that is divisible by for all integers .
Question 3
For the statement " is divisible by ", write the base case value and expand under the hypothesis .