Extension Proof – Inductive reasoning

This lesson comprises two (2) master classes focusing on:

• Proof by mathematical induction

Content:

MEX-P2

• prove results using mathematical induction where the initial value of \( n \) is greater than 1, and/or \( n \) does not increase strictly by 1, for example prove that \( n^2+2n \) is a multiple of 8 if n is an even positive integer
• understand and use sigma notation to prove results for sums, for example: \( \sum_{n=1}^{N} \frac{1}{(2n+1)(2n−1)}=\frac{N}{2N+1} \)
• Understand and prove results using mathematical induction, including inequalities and results in algebra, calculus, probability and geometry. For example:
  • prove inequality results, eg \( 2^n>n^2 \), for positive integers \( n>4 \)
  • prove divisibility results, eg \( 3^{2n+4}−2^{2n} \) is divisible by 5 for any positive integer \( n \)
  • prove results in calculus, eg prove that for any positive integer \( n \), \( \frac{d}{dx}(xn)=nx^{n−1} \)
  • prove results related to probability, eg the binomial theorem: \( (x+a)^n= \sum_{r=0}^{n} {}^{n}C_{r} x^{n−r} a^r \)
  • prove geometric results, eg prove that the sum of the exterior angles of an n-sided plane convex polygon is \( 360^\circ \)
• Use mathematical induction to prove first-order recursive formulae