Algebraic proof

The Maths Habit

Pick three whole numbers in a row and add them. What do you notice? Will it always happen?

The total is always a multiple of 3

Trying examples cannot show it always works

Always even, always odd or could be either?

  1. 2n

    Always even

  2. 2n + 1

    Always odd

  3. n + 4

    Could be either

  4. 2n + 6

    Always even

Consecutive
One after another, in order.
Proof
An argument that shows a statement is true for every number, not just for some.

Three numbers in a row: 4, 5 and 6

456

4 + 5 + 6 = 15

Move one square from the longest row to the shortest

555

Three rows of 5, so 15 = 3 × 5, a multiple of 3

Any three in a row: n, n + 1, n + 2

nnnn + 1n + 1n + 1

n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)

The quick way: write the numbers using n, collect, then factorise.

Prove that the sum of four consecutive even numbers is a multiple of 4.

write the numbers using n
2n+2n + 2+2n + 4+2n + 6
collect like terms
=8n + 12
factorise to show it
=4(2n + 3)
So it isa multiple of 4

Prove that the sum of four consecutive odd numbers is a multiple of 8.

write the numbers using n
2n + 1+2n + 3+2n + 5+2n + 7
collect like terms
=8n + 16
factorise to show it
=8(n + 2)
So it isa multiple of 8

Prove that the sum of four consecutive even numbers is a multiple of 4.

write the numbers using n
2n+2n + 2+2n + 4+2n + 6
collect like terms
=8n + 12
factorise to show it
=4(2n + 3)
So it isa multiple of 4

Prove that (n + 5)2 − n2 is a multiple of 5.

expand the brackets
n2 + 10n + 25 − n2
collect like terms
=10n + 25
factorise to show it
=5(2n + 5)
So it isa multiple of 5

Algebraic proof worksheet, page 1

Answer sheet

  1. Prove that the square of an odd number is odd.

    2(2n2 + 2n) + 1

  2. Dan tries three examples and they all work. Has he proved it?

    No: examples are not a proof

  1. Prove that the sum of three consecutive even numbers is a multiple of 6.

    2n + 2n + 2 + 2n + 4 = 6(n + 1)

  2. Prove that (n + 3)2 − n2 is a multiple of 3.

    6n + 9 = 3(2n + 3)

  3. Lucy checks 3 + 4 + 5 and 10 + 11 + 12. She says this proves the sum of three consecutive whole numbers is a multiple of 3. Explain why this is not a proof.

    Only two examples, not every number
Exit tickets to print