Edexcel IAL Mathematics P2.1.2 proof by exhaustion

Questions either use tables of possible positive integers or residue classes such as 3p+1 and 3p+2, so every permitted case must be shown.

Exam points

Complete all table rows for possible a, b and c values before testing abc divisibility.
Split integers into residue cases such as 3p+1 and 3p+2, then simplify each case.
Conclude only after every allowed case gives the required multiple or non-multiple result.

[Maximum number: 4]

Use proof by exhaustion to prove that if n is an integer then

5 n^{2}+n+12

is always even.

For n odd, let n=2k+1:

5n^2+n+12=5(2k+1)^2+2k+1+12

=20k^2+22k+18=2(10k^2+11k+9)

For n even, let n=2k:

5n^2+n+12=5(2k)^2+2k+12

=20k^2+2k+12=2(10k^2+k+6)

Both expressions are even, so $5n^2+n+12$ is even for all integers n.

M1: starts proof by considering odd or even n.
M1: considers both odd and even cases.
A1: both cases attempted with at least one correct even expression.
A1: complete proof with suitable conclusion.

All question bank results loaded

P2.1.2 - Proof by exhaustion question 1