GCSE Mathematics(9 - 1) - Algebraic Proofs

This chapter comes as a bonus for those who read and learn the previous chapter - a great incentive. It mainly focuses on the proof involving even and odd numbers that often appear in GCSE 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 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.

 

 

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