Firstly, choose $n$ and $n+1$ to be any two consecutive integers. Next, take the squares of these integers to get $n^2$ and $(n+1)^2$ 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 $n+n+1=2n+1$. Hence, we have proved by deduction that the difference between the squares of any two consecutive integers is equal to the sum of those integers.

# How to do Proof by Deduction – Examples & Videos

## Proof by Deduction Notes

**Proof by deduction** is a process in maths where we show that a statement is true using well-known mathematical principles. With this in mind, try not to confuse it 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. Furthermore, *deduction* is the noun associated with the verb *deduce*. It follows that **proof by deduction** is the demonstration that something is true by showing that it must be true for all instances that could possibly be considered. Hence, it is not sufficient to check that a statement is true for a few example numbers – this is a mistake that is often made.

In maths,** proof by deduction** usually requires the use of algebraic symbols to represent certain numbers. For this reason, the following are very useful to know when trying to prove a statement by deduction:

- Use $n$ to represent any integer. Use $n$ and $m$ to represent any two integers.
- Consequently, use $n$, $n+1$ and $n+2$ to represent 3 consecutive integers.
- In addition, if $n$ represents any integer, then $2n$ represents any even integer and $2n+1$ represents any odd integer.
- It follows that $2n$ and $2n+2$ represent any two consecutive even numbers. Alternatively, $2n-1$ and $2n+1$ represent any two consecutive odd numbers.
- Furthermore, use $n^2 $ and $m^2$ to represent any two square numbers.
- $n^2$ and $(n+1)^2$ represent any two consecutive square numbers and so on…

Note that a certain amount of algebra is required when completing proofs. For example, expanding $(n+1)^2$ as $(n+1)(n+1)=n^2+2n+1$. See examples and videos for more details.

## Examples of Proof by Deduction

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

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

By completing the square $x^2-4x+9$ can be written as $(x-2)^2+5$. Note that $(x-2)^2$ is positive for any x as it is a square number . Subsequently, adding 5 will retain its positivity.

### Exam-Style Proof Questions

## More on Proof

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