Geometric Sequence Calculator
Calculate anything about a geometric sequence: the nth term, the sum of the first n terms, the infinite sum when it converges, and the sum between any two positions. Switch to Reverse Solve to find the first term and common ratio from two known terms, or to find how many terms are needed to reach a given value.
Formula
Worked example
With a₁ = 2, r = 3, n = 5: a₅ = 2 x 3^4 = 2 x 81 = 162. The sum S₅ = 2 x (1 - 3^5) / (1 - 3) = 242 (that is 2 + 6 + 18 + 54 + 162). Reverse solve: given a₂ = 6 and a₅ = 162, r = (162/6)^(1/3) = 27^(1/3) = 3 and a₁ = 6 / 3^1 = 2.
How a geometric sequence works
A geometric sequence is a list of numbers where each term is found by multiplying the previous one by a fixed value called the common ratio, written r. Starting from the first term a₁, the terms run a₁, a₁r, a₁r², a₁r³ and so on. The explicit formula aₙ = a₁ x r^(n-1) lets you jump straight to the 100th term without listing every step. The recursive form is simply aₙ = a_{n-1} x r, which describes how each term is built from the one before it. These two representations are equivalent: the explicit formula is the recursive one solved out over n steps. Geometric growth appears throughout mathematics and the natural world: compound interest, population models, drug metabolism, radioactive decay, and signal attenuation all follow the same multiplicative rule.
Summing the terms: finite and infinite series
Adding the first n terms uses the closed form Sₙ = a₁(1 - r^n) / (1 - r), which collapses a long addition into a single division. The formula is undefined when r = 1, but in that case every term equals a₁, so the sum is simply n x a₁. A partial sum from term m to term n is Sₙ minus S_{m-1}, letting you isolate any window of consecutive terms. When the ratio sits strictly between -1 and 1, the terms shrink toward zero and the total of all infinitely many terms converges to a finite limit: S∞ = a₁ / (1 - r). That convergence is the mathematics behind repeating decimals, geometric probability, and the famous result that 0.999... equals exactly 1.
Reverse-solve: finding the ratio or position from known terms
Sometimes you know two term values but not the ratio. If you know term a_p and term a_q, the ratio satisfies r^(q-p) = a_q / a_p, so r = (a_q / a_p)^(1/(q-p)). Plug that r back into the term formula to recover a₁. The "find n" mode runs the algebra in the other direction: given a₁, r, and a target term value, it solves a₁ x r^(n-1) = target by taking logarithms, giving n = 1 + log(target/a₁) / log(r), then rounds to the nearest integer. This is useful for questions like "how many years does it take a population growing at 5% per year to triple?" or "after how many half-lives does radioactivity fall below 1%?"
Sequence behavior by common ratio
| Range of r | Behavior | Infinite sum? | Example |
|---|---|---|---|
| r > 1 | Positive, growing | No | r = 2: 1, 2, 4, 8, 16... |
| r = 1 | Constant | No | r = 1: 5, 5, 5, 5... |
| 0 < r < 1 | Positive, decaying | Yes | r = 0.5: 8, 4, 2, 1, 0.5... |
| r = 0 | Zero after first term | No | r = 0: 3, 0, 0, 0... |
| -1 < r < 0 | Alternating, decaying | Yes | r = -0.5: 4, -2, 1, -0.5... |
| r = -1 | Alternating, constant |term| | No | r = -1: 2, -2, 2, -2... |
| r < -1 | Alternating, growing | No | r = -2: 1, -2, 4, -8... |
How the ratio determines whether a geometric sequence grows, shrinks, or oscillates.
Frequently asked questions
What is the common ratio in a geometric sequence?
The common ratio r is the fixed number you multiply each term by to get the next one. Find it by dividing any term by the term immediately before it: in 3, 6, 12, 24 the ratio is 6/3 = 2. The ratio can be any real number except zero.
How is a geometric sequence different from an arithmetic one?
An arithmetic sequence adds a fixed amount each step, producing a straight-line pattern. A geometric sequence multiplies by a fixed ratio each step, producing exponential growth or decay. Interest on a savings account compounds geometrically; a salary raised by a fixed dollar amount grows arithmetically.
When does the sum of a geometric series converge to a finite value?
Only when the common ratio has an absolute value strictly less than 1. In that case S∞ = a₁ / (1 - r). If |r| is 1 or larger the partial sums grow without bound, so no finite total exists.
How do I find the ratio when I only know two terms?
Use the reverse-solve mode: enter the two term values and their positions. The calculator finds r = (a_q / a_p)^(1/(q-p)) and then recovers a₁. For example, if the second term is 6 and the fifth term is 162, r = (162/6)^(1/3) = 3.
What is the partial sum from term m to term n?
It is Sₙ minus S_{m-1}: compute the full sum to n, subtract the sum to position m-1, and the result is the total of only those terms from position m through n. This is useful when you want the sum over a segment of a longer sequence, not from the beginning.
Can the common ratio be negative or fractional?
Yes. A negative ratio produces an alternating sequence that flips sign each step. A fractional ratio between -1 and 1 produces a converging sequence whose infinite sum is finite. The reverse-solve mode handles non-integer ratios correctly as long as the two positions have an integer difference.