# How to find the term to term rule and the position to term rule of a number sequence.

If you have a number sequence then the numbers in the sequence are called the terms. So if you are asked for the term to term rule then you are looking for a rule that takes you from the previous term in the sequence to the next term in the sequence.

For example, if you have a number sequence that goes, 2,5,8,11,14 then the term to term rule is add on 3 (since its increasing by 3 each time). This can be written down more formally as:

**x _{n+1 }= x_{n }+ 3**

x_{n+1 } is the next term in the sequence.

x_{n }is the term that you are currently on

Or if you have another sequence that goes 13,11,9,7,5 then the tem to term rule is take away 2 (since the sequence is decreasing by 2 each time). Again this can be written using as algebra as:

**x _{n+1 }= x_{n }- 2**

Finding the position to term rule for a number sequence is a little harder (also known as the nth term). The first term in the sequence is in position 1, the second term is in position 2, the third term is in position 3, the fourth term is in position 4, and the fifth term is in position 5. So the position to term rule is the rule that you takes from the position values to the term in the sequence.

For example, a sequence goes 5,9,13,17,21... then the position to term rule is to multiply the position number by 4 and add on 1.

Since:

1 × 4 + 1 = 5 (since 5 is the 1^{st} number in the sequence).

2 × 4 + 1= 9 (since 9 is the 2^{nd} number in the sequence).

3 × 4 + 1 = 13 (since 13 is the 3^{rd} number in the sequence).

4 × 4 + 1 = 17 (since 17 is the 4^{th} number in the sequence).

5 × 4 + 1 = 21 (since 21 is the 5^{th} number in the sequence).

The rule can be written using algebra as 4n + 1 (where n is the position number).

Or if your sequence goes 1,8,15,22,29... then the position to term rule is to multiply by 7 and take off 6.

Since:

1 × 7 - 6 = 1 (since 1 is the 1^{st} number in the sequence).

2 × 7 - 6= 8 (since 8 is the 2^{nd} number in the sequence).

3 × 7 - 6 = 15 (since 15 is the 3^{rd} number in the sequence).

4 × 7 - 6 = 22 (since 22 is the 4^{th} number in the sequence).

5 × 7 - 6 = 29 (since 29 is the 5^{th} number in the sequence).

The rule can be written using algebra as 7n – 6

If you need more help on working out the position to term rule then click here:

How to find the nth term of an increasing linear sequence.

## Comments

**Xavier Nathan** from Isle of Man on June 06, 2013:

This hub just deals with Arithmetic Sequences and there is a formula to find the nth term of an A.P. (Arithmetic Progression) which is so easy to use. Tn=a+(n-1)d

"a" is the first term and "d" is the common difference.

There are also Geometric Sequences where you multiply to get from one term to the next e.g. 3, 6, 12, 24, ... The formula to find the nth term where "r" is the common ratio is Tn = ar^n-1

Perhaps you might compare the two types of sequences in your next hub. Thank you for taking the time to write on this topic.