Generating sequences: term-to-term and position-to-term rules
A sequence is an ordered list of numbers (its terms). There are two main ways to define one:
Term-to-term rule
Tells you how to get the next term from the current one. "Start at 3; add 4 each time" → 3, 7, 11, 15, 19, …
To use a term-to-term rule you need a starting term and the rule. Without the start, you can't generate the sequence.
Position-to-term rule (nth term formula)
A formula u_n (or a_n or T_n) gives the term directly from its position.
u_n = 4n - 1 → u_1 = 3, u_2 = 7, u_3 = 11, … (same sequence as above).
The position-to-term rule is more powerful — it lets you compute the 100th term instantly without finding the previous 99.
✦Worked example— Worked example — convert term-to-term to position-to-term
Sequence 5, 8, 11, 14, … is term-to-term "+3 from 5".
The first differences are constant 3, so it's an arithmetic sequence with common difference 3 and first term 5.
u_n = 5 + (n - 1) × 3 = 3n + 2. Check: u_1 = 5 ✓, u_4 = 14 ✓.
✦Worked example— Worked example — generate from a position-to-term rule
u_n = n² + 2: u_1 = 3, u_2 = 6, u_3 = 11, u_4 = 18.
Differences: 3, 5, 7 (not constant — second differences are constant at 2, marking it a quadratic sequence).
✦Worked example— Worked example — recursive (term-to-term) given the first two
Fibonacci: u_1 = 1, u_2 = 1, u_{n+1} = u_n + u_{n-1}. → 1, 1, 2, 3, 5, 8, 13, 21, …
Notation
u_n— the nth term (n is the position, starting at 1).u_{n+1}— the next term (term afteru_n).u_{n-1}— the previous term.
⚠Common mistakes— Common mistakes (examiner traps)
- Confusing position with value. "n = 5" tells you the position; the value is whatever the formula gives.
- Starting position from 0 instead of 1. GCSE almost always uses n starting at 1.
u_1is the first term. - Writing the term-to-term rule with no start. "+3 each time" only generates the sequence given a starting term.
- Misreading recursive notation.
u_{n+1} = u_n + 5means "next = current + 5", not "current = previous + 5" (which is also true; just be sure of the indexing). - Forgetting fractions or negatives in patterns. Geometric sequences halve, third, etc.
➜Try this— Quick check
A sequence has u_1 = 6 and term-to-term rule "subtract 4". Find u_5.
6, 2, -2, -6, -10. So u_5 = -10.
AI-generated · claude-opus-4-7 · v3-deep-algebra