2022 HSC Mathematics Extension 1
Question 12(f):
Mathematical induction
Use mathematical induction to prove that is divisible by 7 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
- Let .
- Base case : , divisible by 7.
- Inductive hypothesis: assume for some integer .
- Consider .
- Write : .
- Thus is divisible by 7. By induction the result holds for all integers .
- NESA: 3 marks correct proof; 2 for inductive step; 1 for 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 is divisible by 7 for by induction (or note , ).
Question 2
Show and , and deduce .
Question 3
Verify and for are divisible by 7.