# Number Sequences

 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?

### Maths Help suggests:

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

Finding and verifying a formula for the nth 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 . . .
The common step length is 3. So the formula will be
3×n   +   something
 This is because, if the step length is the same for all the terms in the sequence, the formula will be of the format step × n   +   something

For the sequence above, the rule 3×n   +   something would give the values
3×1 + something = 3 + something
3×2 + something = 6 + something
3×3 + something = 9 + something
3×4 + something = 12 + something
3×5 + something = 15 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +2
So the formula for the nth term is   3n + 2

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

 n 1 2 3 4 5 . . . Term 26 31 36 41 46 . . .
The common step length is 5. So the formula will be
5×n   +   something

For the sequence above, the rule 5×n   +   something would give the values
5×1 + something = 5 + something
5×2 + something = 10 + something
5×3 + something = 15 + something
5×4 + something = 20 + something
5×5 + something = 25 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +21
So the formula for the nth term is   5n + 21

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

 n 1 2 3 4 5 . . . Term 20 18 16 14 12 . . .
The common step length is -2. So the formula will be
-2×n   +   something

For the sequence above, the rule -2×n   +   something would give the values
-2×1 + something = -2 + something
-2×2 + something = -4 + something
-2×3 + something = -6 + something
-2×4 + something = -8 + something
-2×5 + something = -10 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +22
So the formula for the nth term is   -2n + 22 or written more neatly   22 - 2n

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 on n2   +   something

 n 1 2 3 4 5 . . . Term 4 7 12 19 28 . . .

For the sequence above, the rule n2   +   something would give the values
12 + something = 1 + something
22 + something = 4 + something
32 + something = 9 + something
42 + something = 16 + something
52 + something = 25 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +3
So the formula for the nth term is   n2 + 3

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 n2 - 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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