Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 arithmetic and geometric sequences? Then this Algebra 1 lesson on Arithmetic and Geometric Sequences as Functions explains each idea step by step, with worked examples.
Quick answer: A sequence is a function on the counting numbers. An arithmetic sequence adds a common difference d each step, so aₙ = a₁ + (n − 1)d. A geometric sequence multiplies by a common ratio r, so aₙ = a₁rⁿ⁻¹. Recursive formulas give each term from the previous one; explicit formulas jump straight to any term.
Standards: F-IF.A.3, F-BF.A.2, F-LE.A.2 · Level: Grade 9 Algebra 1
How is a sequence like a function?
Picture a numbered list. The position n goes in, and the term an comes out. Because only whole-number positions exist, a sequence graphs as separate dots rather than a connected line.
How does an arithmetic sequence work?
Every step of an arithmetic sequence is the same jump up or down. That jump is the common difference d.
Recursive: a1 given, an = an − 1 + d.
Explicit: an = a1 + (n − 1)d.
How does a geometric sequence work?
In a geometric sequence each term is the previous one scaled by a fixed factor, the common ratio r.
Recursive: a1 given, an = r · an − 1.
Explicit: an = a1 · rn − 1.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Rule | add d | multiply by r |
| Example | 4, 11, 18, 25 (d = 7) | 5, 15, 45, 135 (r = 3) |
| Graph | points on a line | points on a curve |
| Linked function type | linear | exponential |
- Check the pattern: each term adds 5, so the sequence is arithmetic and d = 5.
- Write the explicit formula: an = 9 + (n − 1)(5) = 5n + 4.
- Substitute n = 20: 5(20) + 4 = 104.
Using n instead of n − 1. The first term has no steps added, so an counts n − 1 steps.
To classify a sequence, test differences first and ratios second. If neither is constant, the sequence is neither arithmetic nor geometric.
- A sequence is a function whose domain is the positive integers.
- Arithmetic: constant difference d; explicit formula aₙ = a₁ + (n − 1)d.
- Geometric: constant ratio r; explicit formula aₙ = a₁rⁿ⁻¹.
- A recursive formula needs a starting value and a rule from the previous term.
- Arithmetic sequences grow like linear functions, geometric ones like exponentials.
Arithmetic and Geometric Sequences as Functions: frequently asked questions
How do you tell arithmetic from geometric?
Subtract consecutive terms. If the differences match, it is arithmetic. If not, divide consecutive terms; matching quotients mean geometric. If neither is constant, it is neither.
What is the difference between explicit and recursive formulas?
A recursive formula builds each term from the one before it and needs the first term. An explicit formula gives the nth term directly, so you can find the 50th term without listing the first 49.
Why is n − 1 in the explicit formulas?
The first term already exists, so reaching term n takes only n − 1 steps. For 9, 14, 19, … the 20th term uses 19 steps of +5: 9 + 19(5) = 104.
Why are sequence graphs separate points?
The domain contains only whole-number positions, since there is no term number 2.5. So the graph shows dots at n = 1, 2, 3, … rather than a continuous line.
How do you find d or r from two terms that are not next to each other?
For arithmetic, divide the change in value by the number of steps. If a₃ = 17 and a₇ = 37, then d = (37 − 17) ÷ 4 = 5. For geometric, take a root of the quotient.
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