Recursive formulas for sequences
Recursive Formulas for Sequences
Recursive Formulas for Sequences is the opening topic of Sequences & Series in the Common Core State Standards. It is aligned to Standard F-IF.3, which requires students to write sequences both recursively and explicitly.
A recursive formula defines each term of a sequence from the previous one(s) together with a starting value, as in the Fibonacci rule.
Theory
A recursive formula defines each term of a sequence from the one(s) before it, together with a starting value:
For example \(a_n=a_{n-1}+3\) adds 3 each step; the Fibonacci rule \(a_n=a_{n-1}+a_{n-2}\) sums the previous two.
An explicit formula, by contrast, gives \(a_n\) directly from \(n\) — no earlier terms needed.
Recursive and explicit forms of an arithmetic sequence:
How to use a recursive formula
- Start from the given first term(s).
- Apply the rule repeatedly to build each next term.
- Spot the pattern: constant difference \(\to\) arithmetic, constant ratio \(\to\) geometric.
- Convert to an explicit formula when you need a far-off term.
Add 3 each time.
| a_1,a_2,a_3,a_4 | \(=\) | 2,\ 5,\ 8,\ 11 |
Double each time.
| a_2=6,\ a_3=12,\ a_4 | \(=\) | 24 |
Each term is the sum of the previous two.
| 1,1,2,3,a_5 | \(=\) | 5 |
Constant difference \(3\) \(\Rightarrow\) arithmetic; use \(a_n=a_1+(n-1)d\).
| a_n | \(=\) | 2+(n-1)3=3n-1 |
Common pitfalls
Frequently asked questions
What is a recursive formula?
A rule that defines each term from the previous one(s), along with a starting value, e.g. \(a_n=a_{n-1}+3\).
How is a recursive formula different from an explicit one?
Recursive builds a term from earlier terms; explicit gives \(a_n\) directly from \(n\).
What is the Fibonacci sequence?
The recursive sequence \(a_n=a_{n-1}+a_{n-2}\) with \(a_1=a_2=1\), giving 1, 1, 2, 3, 5, 8, ...
Why convert a recursive formula to explicit?
To jump straight to a distant term without computing every term before it.