Geometric Sequence Calculator (aₙ = a₁·rⁿ⁻¹)

Enter the first term a₁ and common ratio r to compute the nth term, partial sums, and infinite sum (when |r| < 1). Includes a term table and bar chart.

Last updated: 2026-08-27

What is a geometric sequence?

A geometric sequence is a list of numbers in which the ratio between any two consecutive terms stays exactly the same all the way through. That constant ratio is called the common ratio r, and together with the first term a₁ it completely determines every value in the sequence. For example, if a₁ = 2 and r = 3, the sequence unfolds as 2, 6, 18, 54, … — each term simply three times the one before it.

This calculator only needs the first term and the common ratio to work out the general term aₙ = a₁·rⁿ⁻¹ and the partial sum Sₙ for as many terms as you like, laid out in a table and a bar chart. When |r| < 1, it also finds the infinite geometric series S∞ that the running total settles down to as more and more terms are added. It is a quick way to double-check values that quickly become unwieldy to compute by hand.

How to use the geometric sequence calculator

Enter the first term a₁ Type in the starting value of the sequence. Positive numbers, negative numbers, and decimals are all accepted.
Enter the common ratio r Type in the ratio between consecutive terms. r > 1 produces growth, 0 < r < 1 produces decay, and r < 0 produces an oscillating sequence.
Choose how many terms to display Set the number of terms shown in the table and chart. Very large counts may be truncated once the values grow too big to display.
Read aₙ and Sₙ from the table For each n, the table lists the nth term aₙ alongside the running partial sum Sₙ, so you can trace the calculation step by step.
Check the chart and infinite sum The bar chart shows the overall growth or decay pattern at a glance, and when |r| < 1 the converged value of the infinite geometric series S∞ is shown as well.

Tips for getting more out of it

When r > 1 the sequence grows rapidly. When 0 < r < 1 it approaches 0. When r < 0 the terms alternate in sign (oscillating sequence).
When |r| < 1, the infinite sum converges to S∞ = a₁ / (1 − r). For example with a₁ = 1 and r = 1/2, S∞ = 2.
The nth term is aₙ = a₁ · r^(n−1) and the partial sum is Sₙ = a₁(1 − rⁿ) / (1 − r) (when r ≠ 1).
The bar chart shows each term as a bar. For large r the bars grow very quickly, so r values between 1.1 and 1.5 make the growth pattern easiest to see.

When this calculator comes in handy

Checking textbook or homework answers

Enter the same a₁, r, and n you worked out by hand to verify the general term or partial sum. Mistakes tend to stand out quickly once the numbers grow large.

Getting a feel for compound growth

Money that grows at a fixed interest rate follows the exact same structure as a geometric sequence with r > 1. Pair this tool with a compound interest calculator when you need the actual monetary figures.

Getting a feel for half-life style decay

Phenomena like the decay of a radioactive sample or the fading of ink concentration, where a fixed fraction is lost each step, can be approximated with a geometric sequence where 0 < r < 1.

Building intuition for algorithmic complexity

When a recursive process doubles or triples its workload at each step, plugging in the growth factor here makes the sheer speed of geometric growth easy to see numerically.

A rough first look at loans or savings plans

Before working through a detailed interest calculation, this is a fast way to get a feel for how quickly a given ratio pushes a value upward over many steps.

Glossary

First term: The very first value in the sequence, written a₁. Together with the common ratio, it uniquely determines every other term in the sequence.
Common ratio: The constant ratio between any two consecutive terms, written r. It satisfies aₙ₊₁ = aₙ × r for every term in the sequence.
General term: A formula that gives the nth term of the sequence directly in terms of n. For a geometric sequence it is aₙ = a₁ · rⁿ⁻¹.
Partial sum: The sum of the first n terms of the sequence, written Sₙ. It can be computed as Sₙ = a₁(1 − rⁿ) / (1 − r) whenever r ≠ 1.
Infinite geometric series: The sum obtained by adding every term of a geometric sequence forever. It converges to a finite value S∞ = a₁ / (1 − r) only when |r| < 1.
Convergence: The property of a sum or sequence of values approaching a single fixed value ever more closely as more terms are added. A geometric series converges precisely when |r| < 1.
Divergence: The opposite of convergence: the sum keeps growing without bound, or oscillates forever, instead of settling on a single value. A geometric series diverges whenever |r| ≥ 1.

FAQ

The infinite sum S∞ = a₁ / (1 − r) is finite only when |r| < 1. When |r| ≥ 1 the terms do not shrink fast enough and the sum diverges (goes to infinity or oscillates without bound).

A negative common ratio produces an oscillating sequence where terms alternate in sign. For example, a₁ = 1 and r = −2 gives 1, −2, 4, −8, 16, …

Let S = a₁ + a₁r + … + a₁r^(n−1). Multiply both sides by r: rS = a₁r + … + a₁rⁿ. Subtract: S − rS = a₁ − a₁rⁿ, giving S(1−r) = a₁(1−rⁿ), so Sₙ = a₁(1−rⁿ)/(1−r).

Side Note — Geometric growth and the paper-folding myth

A classic illustration of geometric growth: if you fold a 0.1 mm sheet of paper in half repeatedly, after n folds its thickness is 0.1 × 2ⁿ mm. The Moon is about 384,400 km away — roughly 3.844 × 10¹¹ mm — so 42 folds would theoretically reach the Moon (2⁴² ≈ 4.4 × 10¹²).

In practice, paper stiffness limits a standard sheet to about 7–8 folds. In 2012, high-school student Britney Gallivan set a world record by folding a long strip of paper 12 times. The mathematical reality remains: geometric sequences grow far beyond our intuition, which is why compound interest, viral spread, and population growth all follow the same pattern.

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