NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Proof MEX-P2 Further Proof by Mathematical Induction

## Outcomes

#### A student:

• MEX12-1

understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts

• MEX12-2

chooses appropriate strategies to construct arguments and proofs in both practical and abstract settings

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is to use the technique of proof by mathematical induction to prove results in series, divisibility, inequality, calculus and geometry.

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 the nature of inductive proof, including the ‘initial statement’ and the inductive step (ACMSM064)
• understand and describe the variation that is required to prove results that are true for all integers greater than a certain integer (i.e. where the initial value of $$n$$ is a value greater than 1)
• prove results using mathematical induction where the initial value of $$n$$ is greater than 1, for example prove that $$n^2 + 2n$$ is a multiple of 8 if $$n$$ is even
• understand and prove results using mathematical induction, including but not limited to results in series and divisibility tests (ACMSM065, ACMSM066)
• understand and use sigma notation to represent and abbreviate the sum of a series for example: $$\displaystyle{ \sum_{n=1}^3 {\left(10n+2\right)} = 12 + 22 +32 }$$
• prove results for sums, for example, $$\displaystyle{\sum_{n=1}^N \frac 1 {\left(2n+1\right)\left(2n-1\right)} = \frac N {2N+1} }$$
• prove divisibility results, for example, $$3^{2n+4} -2^{2n}$$ is divisible by 5 for any integer $$n$$, $$n \geq 0$$, using proof by induction
• understand and prove results using mathematical induction, including but not limited to inequalities and results in algebra, calculus, probability and geometry
• prove inequality results, for example $$2^n > n^2$$, for $$n >4$$
• prove results for calculus, for example prove that for any positive integer $$n$$, $$\frac d {dx} x^n=nx^{n-1}$$
• prove results related to probability, for example the binomial theorem: $$(x+a)^n = \displaystyle{ \sum_{r=0}^n {}^n C_r \ x^{n-r} a^{r} }$$
• prove geometric results, for example, prove that the number of diagonals of a convex polygon with $$n$$ vertices is equal to $$\frac 1 2 n \left(n-3\right)$$
• use mathematical induction to prove first order recursive formulae
• recognise situations where proof by mathematical induction is not appropriate