Arithmetic vs geometric sequences
Updated on June 30, 2026 · by Rafael Rossi
Arithmetic and geometric sequences are the two simplest ways a series can grow — and the difference between them is the same one that separates saving money from investing it, or a contained outbreak from an uncontrolled one.
The two definitions
- Arithmetic sequence: each term is the previous one plus a constant, called the common difference. 2, 5, 8, 11, 14… (difference 3).
- Geometric sequence: each term is the previous one times a constant, called the common ratio. 2, 6, 18, 54, 162… (ratio 3).
Arithmetic grows in a straight line; geometric grows in a curve that accelerates. It is the same distinction as simple versus compound interest, and as a $200 raise versus a 5% raise.
The formulas
| Arithmetic | Geometric | |
|---|---|---|
| nth term | an = a₁ + (n − 1)·d | an = a₁ · rn−1 |
| Sum of n terms | S = n·(a₁ + an) ÷ 2 | S = a₁·(rn − 1) ÷ (r − 1) |
The arithmetic sum formula has an elegant origin: pair the first term with the last, the second with the second-to-last, and so on — every pair adds to the same value. That is why it reads as "the average of the extremes times the number of terms".
Worked example
Arithmetic: first term 5, difference 4. What is the 20th term?
a₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81. And the sum of 20 terms: 20 × (5 + 81) ÷ 2 = 860.
Geometric: first term 5, ratio 2. What is the 20th term?
a₂₀ = 5 × 219 = 5 × 524,288 = 2,621,440. The sum of 20 terms: 5 × (220 − 1) = 5,242,875.
Same position in the sequence, same starting point: 81 against 2.6 million. That is the difference between adding and multiplying, applied nineteen times.
The chessboard legend
The story is old and the maths is checkable. The inventor of chess asks for one grain of wheat on the first square, two on the second, four on the third, doubling to the 64th. The king finds the request modest.
The sum is a geometric series with ratio 2 and 64 terms: 264 − 1 ≈ 18.4 quintillion grains. At 25 milligrams a grain, that is about 460 billion tonnes — roughly 890 years of current world rice production.
What the legend teaches is not about wheat: it is that human intuition judges geometric growth from its first terms, where it looks harmless. Adding everything up to the 32nd square — half the board — gives "only" 4.3 billion grains, about 107 tonnes. One tanker truck. The entire absurdity happens in the second half.
Where each one shows up in real life
- Arithmetic: straight-line depreciation, a fixed-dollar annual raise, seat counts per row in a theatre, a constant-principal repayment schedule.
- Geometric: compound interest, population growth, disease transmission, a drug's half-life (ratio below 1), accumulated inflation, a car's percentage depreciation.
When a geometric series has a finite sum
A result that surprises people: if the ratio lies between −1 and 1, the infinite geometric series has a finite sum, given by S = a₁ ÷ (1 − r).
Adding 1 + 1/2 + 1/4 + 1/8 + … forever gives exactly 2. That is the mathematical resolution of Zeno's paradox: infinitely many steps, each half the previous, fit inside a finite distance. And it is the same formula that explains why 0.999… = 1.
Frequently asked questions
How do I tell which type a sequence is? Subtract consecutive terms: a constant difference means arithmetic. Divide consecutive terms: a constant quotient means geometric. If neither, it is something else.
Can the difference or ratio be negative? In both cases yes. A negative difference makes the sequence decrease in a straight line; a negative ratio makes terms alternate in sign (2, −6, 18, −54…).
Can a geometric sequence contain zero? Negative terms, yes. Zero, no: if any term were zero, every following term would be zero and the ratio would cease to exist.
Why does this matter outside an exam? Because almost every financial decision is a choice between the two. Putting $500 a month under the mattress is arithmetic. The same $500 earning compound interest is geometric — and over thirty years, the gap between them exceeds everything you ever deposited.