Arithmetic vs Geometric Sequences
Arithmetic sequences add the same number each time; geometric sequences multiply by it. The formulas, how to tell them apart, and worked examples.
An arithmetic sequence adds the same number every time to get the next term; a geometric sequence multiplies by the same number every time instead. That one difference — add or multiply — changes everything about how fast the sequence grows and which formula you use to find any term.
Arithmetic sequences: add a common difference
In an arithmetic sequence, each term is the one before it plus a fixed number called the common difference, d.
2, 5, 8, 11, 14, … (d = 3)
The nth term formula is:
This is the arithmetic sequence formula: aₙ = a₁ + (n − 1)d, where a₁ is the first term. For the sequence above, the 10th term is a₁₀ = 2 + (10 − 1)(3) = 2 + 27 = 29. Arithmetic sequences grow (or shrink, if d is negative) at a steady, constant rate — plot the terms against their position and you get a straight line.
Geometric sequences: multiply by a common ratio
In a geometric sequence, each term is the one before it multiplied by a fixed number called the common ratio, r.
2, 6, 18, 54, 162, … (r = 3)
The nth term formula is:
This is the geometric sequence formula: aₙ = a₁ × r^(n − 1). For the sequence above, the 6th term is a₆ = 2 × 3⁵ = 2 × 243 = 486. Geometric sequences grow (or shrink) multiplicatively — plot them and you get a curve that shoots up fast (or decays toward zero if |r| < 1), not a straight line.
How to tell which one you have
- Subtract consecutive terms. If the difference is the same every time, it's arithmetic. 2, 5, 8, 11 → differences are 3, 3, 3: arithmetic.
- If the differences aren't constant, divide consecutive terms instead. If the ratio is the same every time, it's geometric. 2, 6, 18, 54 → ratios are 3, 3, 3: geometric.
- If neither is constant, it isn't a simple arithmetic or geometric sequence — it might follow a different rule entirely, like squaring or a recursive pattern (e.g. Fibonacci).
A worked comparison
Start both sequences at 2, and compare adding 3 each time with multiplying by 3 each time:
| Term | Arithmetic (add 3) | Geometric (× 3) |
|---|---|---|
| 1st | 2 | 2 |
| 2nd | 5 | 6 |
| 3rd | 8 | 18 |
| 4th | 11 | 54 |
| 5th | 14 | 162 |
By the 5th term the geometric sequence is already more than 10 times bigger than the arithmetic one, even though they started the same and used the same number (3). That gap only widens — this is the same reason compound interest (geometric) eventually outgrows simple interest (arithmetic) on the same rate.
Frequently asked questions
Can a sequence be both arithmetic and geometric?
Only if it's constant — every term the same number. Adding 0 each time and multiplying by 1 each time both leave the sequence unchanged, so a constant sequence technically satisfies both definitions. Any sequence that actually changes can only be one or the other, not both.
What is the common ratio in a geometric sequence?
The common ratio is the fixed number each term is multiplied by to get the next one. Find it by dividing any term by the one before it: in 3, 12, 48, 192, the common ratio is 12 ÷ 3 = 4 (check: 48 ÷ 12 = 4 too).
Why do geometric sequences grow faster than arithmetic ones?
Arithmetic growth adds a fixed amount regardless of how big the sequence already is. Geometric growth multiplies, so the amount added each step grows along with the sequence itself — a bigger number multiplied by the same ratio gives a bigger jump than the last one, which compounds over time.
Try it in the Sequences tool
The Sequences tool lets you build either type term by term, watch the pattern plotted on a graph, and see the nth-term formula update live as you change the starting value and the common difference or ratio. Check every formula, including both nth-term rules, on the Maths formulas page, or practise reading a straight-line pattern off real data in how to find g with a simple pendulum.
See it for yourself, in 3D and free.
Try it in the Sequences & Series