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Sequences and the nth term
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A sequence is a list of numbers that follows a rule. Spotting the rule lets you predict any term.
Term-to-term rules
A term-to-term rule tells you how to get from one term to the next.
Start at 3 and add 5: 3, 8, 13, 18, ...
Start at 3, double and add 1: 3, 7, 15, 31, ...
Start at 3 and add 5: 3, 8, 13, 18, ...
Start at 3, double and add 1: 3, 7, 15, 31, ...
Position-to-term rules: the nth term
The nth term gives any term from its position n.
nth term 3n + 2: n = 1 gives 5, n = 2 gives 8, n = 10 gives 32.
To find it for an arithmetic sequence (same difference each time): difference × n, then adjust.
6, 10, 14, 18: difference 4, so 4n gives 4, 8, 12, 16. Add 2: 4n + 2.
nth term 3n + 2: n = 1 gives 5, n = 2 gives 8, n = 10 gives 32.
To find it for an arithmetic sequence (same difference each time): difference × n, then adjust.
6, 10, 14, 18: difference 4, so 4n gives 4, 8, 12, 16. Add 2: 4n + 2.
Other sequences
Geometric: multiply by the same number each time: 2, 6, 18, 54 (× 3).
Square numbers: 1, 4, 9, 16, 25.
Triangular numbers: 1, 3, 6, 10, 15.
Fibonacci: add the previous two: 1, 1, 2, 3, 5, 8.
Square numbers: 1, 4, 9, 16, 25.
Triangular numbers: 1, 3, 6, 10, 15.
Fibonacci: add the previous two: 1, 1, 2, 3, 5, 8.
Find the nth term of 5, 8, 11, 14, ...
- The difference is 3, so start with 3n: 3, 6, 9, 12
- Each term is 2 more than 3n
Answer: 3n + 2
Term-to-term rules go from one term to the next. The nth term jumps straight to any position: for an arithmetic sequence, use (difference)n then adjust.
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