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.

 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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Recommended Reading

 

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

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