A geometric sequence multiplies by the same factor at every step: 2, 6, 18, 54, … triples each time. Geometric patterns model compound growth and decay, from interest and population to drug clearance, depreciation and bouncing balls. Enter the first term and common ratio, or any two terms and their positions, to get the nth term, the sum of the first n terms, the infinite sum when it exists, both formulas and a table of terms. Very large terms are shown in scientific notation instead of overflowing.
How to use the geometric sequence calculator
- Choose whether you know the first term and ratio or any two terms.
- Enter the values. For two terms, give each position and value.
- Enter the term number n to find; the calculator also sums the first n terms.
- Read aₙ, Sₙ and, when |r| < 1, the infinite sum S∞.
Geometric sequence formulas
From two terms at positions p and q: rq − p = aq ÷ ap, then a₁ = ap ÷ rp − 1.
Worked examples
Growth. For 2, 6, 18, … (a₁ = 2, r = 3):
- a₈ = 2 × 3⁷ = 2 × 2,187 = 4,374.
- S₈ = 2 × (1 − 3⁸) ÷ (1 − 3) = 2 × (−6,560) ÷ (−2) = 6,560.
- Since r > 1, the infinite series diverges.
From two terms. If the 2nd term is 6 and the 5th is 162, then r³ = 162 ÷ 6 = 27, so r = 3 and a₁ = 6 ÷ 3 = 2: the same sequence.
Decay. A ball dropped from 100 cm rebounds to half its height each time: 100, 50, 25, … (r = 0.5). The 6th height is 100 × 0.5⁵ = 3.125 cm, and the first six heights total 196.875 cm. The heights can never add up to more than S∞ = 100 ÷ (1 − 0.5) = 200 cm.
Understanding geometric growth
The ratio decides everything
| Ratio | Behavior |
|---|---|
| r > 1 | terms grow without bound (exponential growth) |
| 0 < r < 1 | terms shrink toward 0 (exponential decay) |
| −1 < r < 0 | terms shrink while alternating in sign |
| r < −1 | terms alternate and grow in size |
| r = 1 or r = −1 | terms repeat or flip between two values |
Why the sum formula works
Multiply Sₙ = a₁ + a₁r + … + a₁rn−1 by r and subtract: every middle term cancels, leaving Sₙ − rSₙ = a₁ − a₁rn. Dividing by 1 − r gives the formula. When |r| < 1, rn vanishes as n grows, which is where the infinite-sum formula comes from. The repeating decimal 0.999… is the geometric series 9/10 + 9/100 + …, with a₁ = 0.9 and r = 0.1, so it equals 0.9 ÷ 0.9 = 1 exactly.
Doubling and halving
A ratio of 2 doubles each step: after 10 steps a quantity is 1,024 times larger, after 20 steps over a million times. That is why exponential processes surprise people. The same arithmetic in reverse gives half-lives.
Related tools
For constant differences instead of constant ratios, use the arithmetic sequence calculator. Money growing by a fixed percentage is handled with deposits and compounding options by the compound interest calculator. For powers reduced modulo a number, as in cryptography, see the modular exponentiation calculator.
Frequently asked questions
What is a geometric sequence?
A sequence in which each term is the previous term multiplied by the same number, the common ratio r. For example 2, 6, 18, 54, … has r = 3, and 100, 50, 25, … has r = 0.5. A negative ratio makes the signs alternate.
When does an infinite geometric series have a sum?
Only when the ratio is strictly between −1 and 1. Then the terms shrink toward zero fast enough that the total settles at S∞ = a₁ ÷ (1 − r). For |r| ≥ 1 the terms do not shrink, and the series diverges (or, for r = −1, swings back and forth forever).
How do I find the ratio from two terms that are not next to each other?
Divide the later term by the earlier one and take the root matching the gap in positions: r = (a_q ÷ a_p)^(1/(q − p)). If the 2nd term is 6 and the 5th is 162, r³ = 27, so r = 3. When the gap is even, both a positive and a negative ratio fit; the calculator reports the positive one and notes the other.
How is a geometric sequence related to compound interest?
A balance growing at a fixed rate each period is a geometric sequence with ratio 1 + rate. $1,000 at 5% a year gives 1,000, 1,050, 1,102.50, … , so after n years the balance is 1,000 × 1.05ⁿ.
Can the common ratio be zero?
Technically r = 0 makes every term after the first equal to zero, which is usually excluded from the definition because the pattern carries no information. The first term must also be nonzero for the ratio to be found from two terms.