From pattern to sequence
A pattern is a rule; a sequence is the ordered list of terms that the rule produces. Once you recognise the pattern, you can describe it in words, numbers, or an algebraic expression. This lets you jump straight to any term without listing them all.
Key idea / 要点
For a sequence that adds the same number each time, the nth term has the form an + b, where a is the common difference.
Building the general term
Take 3, 5, 7, 9, ... The common difference is 2, so start with 2n. When n = 1, 2n = 2, but the first term is 3, so add 1. The rule is 2n + 1. Check: n = 4 gives 2(4) + 1 = 9. Correct.
Worked example / 例题
Find the 10th term of 4, 7, 10, 13, ...
Common difference = 3, so the n th term is 3n + 1 (since 3(1) + 1 = 4).
10th term = 3(10) + 1 = 31.
Describing patterns in words
You can also describe a sequence in words, for example "start at 4 and add 3 each time." Both the word rule and the algebraic rule 3n + 1 describe the same sequence. Using algebra is faster for large positions, such as the 50th term: 3(50) + 1 = 151.
Remember / 记住
- Find the common difference first — that is the coefficient of n.
- Adjust with a constant so the first term matches.
- Always check your rule with a known term.