## Proof by Deduction

As well as subtract, the word deduce means to 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. This usually involves choosing algebraic symbols to represent certain numbers. The following are very useful to know when trying to prove by deduction:

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

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

Choose $n$ and $n+1$ to be any two consecutive integers. The squares of these integers is given by $n^2$ and $(n+1)^2$ respectively where $(n+1)^2=(n+1)(n+1)=n^2+2n+1$. The difference between these numbers is $n^2+2n+1-n^2=2n+1$. Adding together the original two consecutive numbers also gives $n+n+1=2n+1$ and so we have proved by deduction that  the difference between the squares of any two consecutive integers is equal to the sum of those integers.