# Proof by Deduction

Proof by deduction is a process in maths where a statement is proved to be true based on well-known mathematical principles. It is not to be confused with Proof by Induction or Proof by Exhaustion.

The word deduce means to establish facts through reasoning or make conclusions about a particular instance by referring to a general rule or principle. Deduction is the noun associated with the verb deduce. In maths, proof by deduction means that you can prove that something is true by showing that it must be true for all cases that could possibly be considered.

Proof by deduction may require the use of algebraic symbols to represent certain numbers. For this reason, the following are very useful to know when trying to prove by deduction:

• $n$ can represent any number
• Consequently, if $n$ represents any integer, then $2n$ represents any even integer and $2n+1$ represents any odd integer
• In addition, $n$$n +1$,  $n+2$ can be used to represent 3 consecutive integers
• Furthermore, $n^2$ and $m^2$ could be used to represent any two square numbers and so on……

## Example 1

Prove that the difference between the squares of any two consecutive integers is equal to the sum of those integers.

## Example 2

Prove that $x^2-4x+9$ is always positive.

## Sample Exam Question

– as supplied by Edexcel Sample Assessment Material