Algebraic Proof (AQA GCSE Maths)

, for example, where is an integer, is an expression for an even integer.

Other letters could also be used.

, for example, where is an integer, is an expression for an odd integer.

Another possibility is .

, for example, where is an integer, is an expression for an integer that is a multiple of 7.

and , for example, where is an integer, are expressions for two numbers that are consecutive.

Another possibility is and .

and , for example, where and are different integers, are expressions for any two even numbers.

It is not and , as this would mean two consecutive even numbers.

The first line of algebra when proving that the sum of two consecutive integers is odd is:

This shows an integer, , being summed with the next (consecutive) integer, .

True.

If is an integer, then is odd.

It has the form of an odd number, , as is an integer.

False.

It is true that even numbers are written in the form but the part inside the brackets must be an integer.

The problem with is that the part inside the brackets has a in it, so would never be an integer.

Alternatively, you could expand the brackets to see that which is odd (assuming is an integer).

and are two consecutive multiples of 3.

and are squares of two consecutive multiples of 3.

is the difference between the squares of two consecutive multiples of 3.

So the answer is .

Note also how you usually write the bigger number subtract the smaller number, so that the overall result is positive.

Odd numbers have the form .

One way show that is odd is to rewrite as then factorise out a 2 to get .

This now has the form of an odd number as given above.

If you factorised 4 out instead to get , you would need an extra line of working to explain how to get to the form .