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NSW Syllabuses

Mathematics Extension 1 Stage 6 - Year 12 Extension 1 - Proof ME-P1 Introduction to Proof by Mathematical Induction

Outcomes

A student:

  • ME12-1

    applies techniques involving proof or calculus to model and solve problems

  • ME12-6

    chooses and uses appropriate technology to solve problems in a range of contexts

  • ME12-7

    evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms

Subtopic Focus

The principal focus of this subtopic is to explore the use of the technique of proof by mathematical induction to prove results.

Students develop the use of formal mathematical language across various topics of mathematics to prove the validity of given situations using inductive reasoning.

Content

  • Students:
  • understand and use the basic principles of mathematical induction
  • understand the nature of inductive proof, including the ‘initial statement’ and the inductive step (ACMSM064)
  • identify errors in false ‘proofs by induction’, such as cases where only one of the required two steps of a proof by induction is true, and understand that this means that the statement has not been proved
  • prove results using mathematical induction relating to series (ACMSM065, ACMSM066) cct
  • prove results for sums, for example \( 1 + 4 + 9 + \cdots + n^2 = \frac {n \left(n+1\right) \left(2n+1\right) } 6 \) for any positive integer \(n\)
  • prove results using mathematical induction relating to divisibility cct
  • prove divisibility results, for example, \(3 ^{2n} - 1\) is divisible by 8 for any positive integer \(n \geq 0 \)
  • recognise situations where proof by mathematical induction is not appropriate cct