Algebraic Proofs

These worked example mainly focus on the proof involving even and odd numbers that often appear in GCSE/iGCSE examination papers.

In this chapter, an even number is denoted as 2n and an odd number as 2n + 1, where n is an integer.

E.g.1

Show that the sum of squares of two consecutive odd numbers is a multiple of 4 added to 2.
Let the numbers be (2n + 1) and (2n + 3).
(2n + 1)2 + (2n + 3)2 = 4n2 + 4n + 1 + 4n2 + 12n + 9
8n2 + 16n + 10 = 4[n2 + 4n + 2] + 2

E.g.2

Show that the sum of squares of two consecutive even numbers is a multiple of 4.
Let the numbers be (2n) and (2n + 2).
(2n)2 + (2n + 2)2 = 4n2 + 4n2 + 8n + 4
8n2 + 8n + 4 = 4[2n2 + 2n + 1]

E.g.3

Show that the difference of squares of two consecutive numbers is always an odd number.
Let the numbers be x and (x + 1).
(x + 1)2 - x2 = x2 + 2x + 1 - x2
= 2x + 1, an odd number regardless of x.

E.g.4

Show that the difference of squares of two consecutive odd numbers is a multiple of 8.
Let the numbers be (2n + 1) and (2n + 3).
(2n + 3)2 - (2n + 1)2 = 4n2 + 12n + 9 - 4n2 - 4n - 1
8n2 + 8n + 8 = 8[n2 + n + 1]

E.g.5

Show that the expression, x2 - 6x + 11, is always positive regardless of x.
By completing the square, x2 - 6x + 11 = (x - 3)2 + 2
Since, being a square, (x - 3)2 is always positive, so is (x - 3)2 + 2.

E.g.6

Show that the sum of four consecutive integers is always even.
Let the numbers be x, x + 1, x + 2 and x + 3
The sum = x + x + 1 + x + 2 + x + 3 = 4x + 6 = 2(2x + 3), an even number.