Maths · Algebra Foundations
Sequences & nth term
Recognise linear sequences and find nth-term rules.
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What is a linear sequence?
A linear sequence is a pattern of numbers where the same amount is added or subtracted each time. The difference between consecutive terms (one after another) stays constant. For example: 3, 7, 11, 15, 19 increases by 4 each time.
Finding the common difference
The common difference is the amount added (or subtracted) between terms. Subtract any term from the next term to find it. In the sequence 5, 8, 11, 14, the common difference is 8 − 5 = 3.
The nth-term rule
The nth-term rule (or formula) lets you find any term without listing them all. It has the form an + b, where a is the common difference and b is found using the first term. The letter n represents the position number.
Working out the nth-term formula
First find the common difference (a). Then multiply a by n to get an. Compare an with your sequence to find what to add or subtract (b). Check your formula works for the first few terms.
Using the nth-term rule
Once you have the nth-term rule, substitute any position number for n. For example, if the rule is 4n + 1, the 10th term is 4(10) + 1 = 41. This saves time for large position numbers.
Worked examples
1. Find the nth-term rule for the sequence 5, 9, 13, 17, 21...
- Find the common difference: 9 − 5 = 4.
- The 4n sequence is 4, 8, 12, 16, 20...
- Compare with the original: each term is 1 more than 4n.
- The nth-term rule is 4n + 1.
2. Find the 20th term of the sequence with nth-term rule 3n − 2.
- Substitute n = 20 into the formula.
- 3(20) − 2 = 60 − 2.
- The 20th term is 58.
3. Find the nth-term rule for 10, 7, 4, 1, −2...
- Find the common difference: 7 − 10 = −3.
- The −3n sequence is −3, −6, −9, −12, −15...
- Compare: the original is 13 more than −3n.
- The nth-term rule is −3n + 13 (or 13 − 3n).
Try it yourself
Find the nth-term rule for the sequence 6, 11, 16, 21, 26...
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