- From: Mel Sharpe
- Date: 2 April 1999
*Subject: Finding the general term of a sequence*
Please help! For my GCSE coursework I have got to find the formula for a sequence of numbers. I can spot the pattern and explain it in words, but how do you get a formula? |

This is a common problem, and regularly crops up in Investigations as part of GCSE coursework.

Finding and verifying a formula for the n^{th} term of a sequence is the "icing on the
cake" of an investigation. We shall give some examples of how to go about it.

**If the numbers in the sequence increase in EQUAL STEPS** then things are fairly
straightforward. For example:

5 , 8 , 11 , 14 , 17 , ...(step length 3)

26 , 31 , 36 , 41 , 46 , ...(step length 5)

20 , 18 , 16 , 14 , 12 , ...(step length -2)

Let's do the first of these examples.

First, draw up a table, giving each term its "counter" (generally called **n**):

n | 1 | 2 | 3 | 4 | 5 | . . . |

Term | 5 | 8 | 11 | 14 | 17 | . . . |

This is because, if the step length is the same for
all the terms in the sequence, the formula will be of the formatstep × n + something |

For the sequence above, the rule **3×n + something**
would give the values

3×1 +

3×2 +

3×3 +

3×4 +

3×5 +

Compare these values with the ones in the actual sequence - it should be obvious that the value of the

Now let's do the second example....

n | 1 | 2 | 3 | 4 | 5 | . . . |

Term | 26 | 31 | 36 | 41 | 46 | . . . |

For the sequence above, the rule **5×n + something**
would give the values

5×1 +

5×2 +

5×3 +

5×4 +

5×5 +

Compare these values with the ones in the actual sequence - it should be obvious that the value of the

Now let's do the third example....

n | 1 | 2 | 3 | 4 | 5 | . . . |

Term | 20 | 18 | 16 | 14 | 12 | . . . |

For the sequence above, the rule **-2×n + something**
would give the values

-2×1 +

-2×2 +

-2×3 +

-2×4 +

-2×5 +

Compare these values with the ones in the actual sequence - it should be obvious that the value of the

** If the terms do NOT increase in equal steps, then you have to think about things a bit more.**
Here is an example:

4 , 7 , 12 , 19 , 28 , . . . .steps are +3, +5, +7, +9

At GCSE level, if the steps between the terms are not equal, try a rule based onn
^{2} + something |

n | 1 | 2 | 3 | 4 | 5 | . . . |

Term | 4 | 7 | 12 | 19 | 28 | . . . |

For the sequence above, the rule **n ^{2} + something**
would give the values

1

2

3

4

5

Compare these values with the ones in the actual sequence - it should be obvious that the value of the

This is a topic where practice will help you to see the patterns and rules more easily. It is
possible to practise this with a friend. Each think of a formula (eg. *4n + 3* , or
*n ^{2} - 1*) and write down the first four or five terms using the formula.
Then swap your paper with your friend, and try to work out each other's formula.

**To see another question which deals with more complicated number sequences, click
here.**

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