Algebra 2 · Unit 9 · Topic 9.1
Explicit and Recursive Sequence Rules
An explicit rule gives a term directly from its index; a recursive rule gives it from earlier terms and requires enough starting values.
Definition and explanation
An explicit rule gives a term directly from its index; a recursive rule gives it from earlier terms and requires enough starting values. State where indexing begins before interpreting either rule.
Worked example
Let a₁ = 7 and aₙ = aₙ₋₁ + 5 for n ≥ 2. The first terms are 7, 12, 17, and 22, and the explicit rule is aₙ = 7 + 5(n − 1). It gives a₈ = 42 without listing all intermediate terms.
Common mistake
A recursive formula without a starting value does not uniquely determine its terms.
Key takeaway
An explicit rule gives a term directly from its index; a recursive rule gives it from earlier terms and requires enough starting values.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.