It is possible to determine a formula for linear sequences, i.e. sequences where the difference between successive terms is always the same.

The first differences for the number pattern

11 | 14 | 17 | 20 | 23 | 26 | ... |

are | 3 | 3 | 3 | 3 | 3 |

If we look at the sequence 3*n*, i.e. the multiples of 3, and compare it with our original sequence

our sequence | 11 | 14 | 17 | 20 | 23 | 26 |
---|---|---|---|---|---|---|

sequence 3n | 3 | 6 | 9 | 12 | 15 | 18 |

we can see easily that the formula that generates our number pattern is

i.e. u_n = 3n + 8

If, however, we had started with the sequence

38 | 41 | 44 | 47 | 50 | 53 | ... |

the first differences would still have been 3 and the comparison of this sequence with the sequence 3*n*

our sequence | 38 | 41 | 44 | 47 | 50 | 53 |
---|---|---|---|---|---|---|

sequence 3n | 3 | 6 | 9 | 12 | 15 | 18 |

would have led to the formula u_n = 3n + 35.

In the same way, the sequence

–7 | –4 | –1 | 2 | 5 | 8 | ... |

also has first differences 3 and the comparison

our sequence | – 7 | – 4 | – 1 | 2 | 5 | 8 |
---|---|---|---|---|---|---|

sequence 3n | 3 | 6 | 9 | 12 | 15 | 18 |

yields the formula u_n = 3n – 10.

From these examples, we can see that any sequence with constant first difference 3 has the formula

u_n = 3n +where the adjustment constant *c* may be either positive or negative.

This approach can be applied to any linear sequence, giving us the general rule that:

If the *first* difference between *successive* terms is *d*, then

u_n = *d* × n + *c*

Determine a formula for this sequence:

7, | 13, | 19, | 25, | 31, | ... |

First consider the differences between the terms,

7 | 13 | 19 | 25 | 31 | ... |

6 | 6 | 6 | 6 |

As the difference is always 6, we can write,

u_n = 6n + *c*

As the first term is 7, we can write down the equation:

7 | = 6 × 1 + c |

= 6 + c | |

c | = 1 |

So the formula will be,

u_n = 6n + 1

We can check that this formula is correct by testing it on other terms, for example,

the 4th term = 6 × 4 + 1 = 25

which is correct.

Determine a formula for this sequence:

2, | 7, | 12, | 17, | 22, | 27, | ... |

First consider the differences between the terms,

2 | 7 | 12 | 17 | 22 | 27 | ... |

5 | 5 | 5 | 5 | 5 |

The difference between each term is always 5, so the formula will be,

u_n = 5n + *c*

The first term can be used to form an equation to determine *c*:

2 | = 5 × 1 + c |

2 | = 5 + c |

c | = –3 |

So the formula will be,

u_n = 5n – 3

Note that the constant term, *c*, is given by

*c* = first term – first difference

Determine a formula for the sequence:

28, 25, 22, 19, 16, 13, ...

First consider the differences between the terms,

28 | 25 | 22 | 19 | 16 | 13 | ... |

–3 | –3 | –3 | –3 | –3 |

Here the difference is *negative* because the terms are becoming smaller.

Using the difference as –3 gives,

u_n = –3n + *c*

The first term is 28, so

28 | = –3 × 1 + c |

28 | = –3 + c |

c | = 31 |

The general formula is then,

u_n = –3n + 31

or

u_n = 31 – 3n