Sequences and rules
- A sequence is a list of numbers that follows a rule. The term-to-term rule tells you how to get from one term to the next.
- In an arithmetic (linear) sequence, the same amount is added or subtracted each time: 3, 7, 11, 15 (+4).
- In a geometric sequence, each term is multiplied by the same number: 2, 6, 18, 54 (×3).
Special sequences
- Square numbers: 1, 4, 9, 16, 25… Cube numbers: 1, 8, 27, 64…
- Triangular numbers: 1, 3, 6, 10, 15… (add 2, then 3, then 4…).
- Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13… (add the two terms before).
The nth term
- The nth term is a position-to-term rule. For 3, 7, 11, 15, the difference is 4, so start with 4n; 4 × 1 = 4, and the first term is 3, so subtract 1: the nth term is 4n − 1.
- Use it to find any term: the 20th term is 4 × 20 − 1 = 79.
- To check whether a number is in the sequence, solve 4n − 1 = 99: n = 25, a whole number, so 99 is the 25th term.
Key terms
- Sequence
- A list of numbers that follows a rule.
- Term
- One number in a sequence.
- Term-to-term rule
- How to get from one term to the next.
- Arithmetic sequence
- A sequence that goes up or down by the same amount each time.
- Geometric sequence
- A sequence where each term is multiplied by the same number.
- nth term
- A rule that gives any term from its position, such as 4n − 1.
- Common difference
- The amount added each time in an arithmetic sequence.
- Fibonacci sequence
- A sequence where each term is the sum of the two before it.