We can use algebraic proof to show the truth of many mathematical statements.
Prove that (n + 4)2 - (n + 2)2 is always a multiple of 4.
Expand the brackets and simplify:
n2 + 4n + 4n +16 - (n2 +2n + 2n + 4)
= n2 + 4n + 4n + 16 - n2 - 2n - 2n - 4
= 4n + 12
Factorise to show that 4 is a factor and so that the expression is a multiple of 4:
4(n + 3)
We use a set of expression to show different kinds of numbers. These help us with writing algebraic proofs.
|2n||an even number|
|2n + 1||an odd number|
|n, n+1, n+2||consecutive numbers|
|2n, 2n+2, 2n+4||consecutive even numbers|
|2n+1, 2n+3, 2n+5||consecutive odd numbers|
|6n||multiple of 6|
|5n||multiple of 5|
Have a look at this question:
Prove that the sum of any three consecutive even numbers is always a multiple of 6.
Sum means adding. We are going to add three consecutive even numbers (2n, 2n+2, 2n+4):
2n + 2n + 2 + 2n + 4
6n + 6
Factorise to show 6 is a factor:
6(n + 1)