We can use algebraic proof to show the truth of many mathematical statements.

**Prove that (n + 4) ^{2} - (n + 2)^{2} is always a multiple of 4.**

Expand the brackets and simplify:

n^{2} + 4n + 4n +16 - (n^{2} +2n + 2n + 4)

= n^{2} + 4n + 4n + 16 - n^{2} - 2n - 2n - 4

= 4n + 12

Factorise to show that 4 is a factor and so that the expression is a multiple of 4:

4(n + 3)

We use a set of expression to show different kinds of numbers. These help us with writing algebraic proofs.

2n | an even number |

2n + 1 | an odd number |

n, n+1, n+2 | consecutive numbers |

2n, 2n+2, 2n+4 | consecutive even numbers |

2n+1, 2n+3, 2n+5 | consecutive odd numbers |

6n | multiple of 6 |

5n | multiple of 5 |

Have a look at this question:

Prove that the **sum** of any __three consecutive even numbers__ is always a multiple of 6.

**Sum **means adding. We are going to add three consecutive even numbers (2n, 2n+2, 2n+4):

2n + 2n + 2 + 2n + 4

Simplify:

6n + 6

Factorise to show 6 is a factor:

6(n + 1)