A geometric sequence is a list of numbers where each term is the previous one multiplied by a fixed number called the common ratio. That single rule produces everything from doubling patterns to halving patterns to the decimal expansion of a repeating fraction. This calculator returns any term you ask for, the sum of the first n terms, and the infinite sum when the sequence converges — plus the ratio itself when you have two terms and need to work backwards.
Arb Digital built this page as the multiplicative counterpart to our arithmetic sequence calculator, and the distinction between them is the whole point: an arithmetic sequence adds a constant difference each step, while a geometric one multiplies by a constant ratio. Adding produces a straight line; multiplying produces a curve that eventually outruns any straight line no matter how steep.
What This Geometric Sequence Calculator Does
It works in two directions. Given a first term and a common ratio it returns the nth term, the partial sum, the infinite sum where one exists, and the first several terms written out so you can see the pattern. Given a first term and a known later term it solves for the ratio that connects them, which is the form the problem usually takes when the sequence comes from observed data rather than from a textbook.
It also states plainly whether the series converges. That check is not decoration: the formula for an infinite geometric sum produces a perfectly plausible-looking number for a ratio outside the convergent range, and printing that number without a warning is one of the more common calculator defects in this corner of mathematics.
How to Use It
- Choose what you know. Either the first term and the ratio, or the first term and a later term whose position you can state.
- Enter the first term. It can be negative or fractional; only zero is excluded, because a sequence starting at zero stays there.
- Enter the ratio or the known term. A ratio between −1 and 1 shrinks the sequence; a negative ratio makes the signs alternate.
- Set the position n. This is both the term returned and the number of terms in the partial sum.
- Check the behaviour flag before using the infinite sum. It is only a real number when the ratio is strictly inside the convergent range.
The Formulas and How They're Calculated
The nth term is aₙ = a₁ · r⁽ⁿ⁻¹⁾. The sum of the first n terms is Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for any r other than 1, and simply n·a₁ when r equals 1. The infinite sum is S∞ = a₁ ÷ (1 − r), and it exists only when the absolute value of r is strictly less than 1. Solving for the ratio from two terms gives r = (aₙ/a₁)^(1/(n−1)).
Work through the default: a₁ = 3, r = 2, n = 8. The eighth term is 3 × 2⁷ = 3 × 128 = 384. The partial sum is 3(1 − 2⁸) ÷ (1 − 2) = 3(1 − 256) ÷ (−1) = 3 × 255 = 765, which you can verify by adding 3 + 6 + 12 + 24 + 48 + 96 + 192 + 384. Since the ratio is 2, the series diverges and there is no infinite sum. Switching to the second mode with a₁ = 3, a₈ = 384 recovers r = (384/3)^(1/7) = 128^(1/7) = 2, closing the loop. Wolfram MathWorld's geometric sequence entry states these definitions in their general form.
Why an Infinite Sum Can Be Finite
Adding infinitely many positive numbers and getting a finite total sounds like a contradiction until you look at the arithmetic. Take a₁ = 100 and r = 0.5. The terms are 100, 50, 25, 12.5, 6.25 and so on, and the partial sums are 100, 150, 175, 187.5, 193.75 — always closer to 200 and never past it. Each term covers exactly half the remaining gap, so the total approaches 200 without ever reaching it. The formula confirms it: 100 ÷ (1 − 0.5) = 200.
The condition is strict. At r = 1 every term is identical and the sum grows without limit. At r = −1 the partial sums oscillate between two values forever and never settle on anything. Beyond ±1 the terms themselves grow, so the sum cannot possibly converge. Inside that range the terms shrink fast enough that their total is bounded, which is the whole content of the convergence test. MathWorld's geometric series entry gives the proof, and MIT OpenCourseWare's 18.01SC Single Variable Calculus course places it in the wider theory of infinite series.
Negative Ratios and Alternating Signs
A negative common ratio flips the sign at every step. With a₁ = 5 and r = −2 the sequence runs 5, −10, 20, −40, 80 — the magnitudes grow geometrically while the signs alternate. Two things follow that catch people out. First, an alternating sequence has no single trend to eyeball, so a chart of it looks like noise even though the rule is perfectly deterministic. Second, the partial sums oscillate around the eventual value rather than approaching it from one side.
That oscillation is useful when the ratio is inside the convergent range. With a₁ = 1 and r = −0.5 the partial sums go 1, 0.5, 0.75, 0.625, 0.6875, closing in on 2/3 from alternate directions. Because each partial sum overshoots and then undershoots, any two consecutive partial sums bracket the true answer — a property that gives a free error bound on a truncated series and is exploited constantly in numerical work.
Geometric Sequences and Repeating Decimals
Every repeating decimal is a geometric series in disguise, and this is the cleanest demonstration of why the infinite sum formula is exact rather than approximate. The decimal 0.3333… is 0.3 + 0.03 + 0.003 + …, a geometric series with a₁ = 0.3 and r = 0.1. The formula gives 0.3 ÷ (1 − 0.1) = 0.3 ÷ 0.9 = 1/3, exactly.
The same method converts any repeating decimal to a fraction. For 0.121212… take a₁ = 0.12 and r = 0.01, giving 0.12 ÷ 0.99 = 12/99 = 4/33. And it settles the old argument about 0.999…: with a₁ = 0.9 and r = 0.1, the sum is 0.9 ÷ 0.9 = 1. Not approaching 1, not almost 1 — equal to 1, because the infinite sum is defined as the limit of the partial sums and that limit is exactly 1. Our decimal to fraction calculator performs the conversion directly when you just need the answer.
Where Geometric Sequences Show Up
Compound growth is geometric by construction: a balance multiplied by the same factor each period is exactly a geometric sequence, with the sum of the terms giving the total of a series of equal payments each growing at that rate. Depreciation on a declining-balance basis is a geometric sequence with a ratio below 1. Musical tuning is geometric, with each semitone multiplying the frequency by the twelfth root of 2, so twelve semitones give a factor of exactly 2 — the octave.
The same structure underlies exponential decay in physics, resolution steps in binary search, and the way a pixel grid doubles at each zoom level. If you are modelling a quantity over continuous time rather than discrete steps, our exponential growth calculator handles the same relationship with time as a continuous variable, and the compound interest calculator covers the money-specific case with contributions and compounding frequency.
Reading a Sequence Off Real Data
Solving for the ratio from two terms takes three seconds and hides a trap worth naming. Any two numbers define a ratio, so the calculation always succeeds and never tells you whether the data is actually geometric. A quantity that went from 3 to 384 across eight terms gives r = 2 whether it grew smoothly by doubling or jumped in a single step.
The test is to compute the ratio between several consecutive pairs. If the ratios agree to within measurement noise, the sequence is genuinely geometric and the model will project sensibly. If they drift steadily — say from 2.4 down to 1.3 — the process is decelerating and no single ratio describes it. That drift is the normal case in real data, and forcing a constant ratio onto it produces projections that are far too high. When the growth pattern is additive rather than multiplicative, the arithmetic sequence calculator is the right model instead.
Arb Digital tests whether a trend is genuinely multiplicative before projecting it forward, rather than fitting a ratio to two convenient points.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Using n instead of n − 1 in the exponent — the first term has no multiplication applied, so the eighth term uses r to the power of seven.
- Applying the infinite sum formula outside the convergent range — it returns a plausible number for any ratio, but the answer is meaningless unless the ratio is strictly between −1 and 1.
- Confusing the sequence with the series — the sequence is the list of terms, the series is their running total, and "sum" always means the second.
- Fitting a ratio from two points and projecting far forward — two points always fit, so check several consecutive ratios before trusting one.
- Treating a negative ratio as an error — alternating signs are a normal geometric sequence, and the sum formula handles them without modification.
Related Free Tools From Arb Digital
Model a constant difference with the arithmetic sequence calculator, average successive growth rates with the geometric mean calculator, project continuous growth with the exponential growth calculator, work with powers directly using the exponent calculator, or convert a repeating decimal with the decimal to fraction calculator. The free online tools hub lists every maths calculator we publish.
Frequently Asked Questions
A list of numbers in which each term is the previous term multiplied by a fixed value called the common ratio, such as 3, 6, 12, 24 where the ratio is two.
Multiply the first term by the common ratio raised to the power of n minus one. The exponent is one less than the position because the first term has no multiplication applied to it yet.
Only when the absolute value of the common ratio is strictly less than one. In that case the sum equals the first term divided by one minus the ratio.
An arithmetic sequence adds a constant difference between terms; a geometric sequence multiplies by a constant ratio. One grows linearly, the other grows or shrinks proportionally.
Yes. A negative ratio makes the signs alternate while the magnitudes still follow the same multiplicative rule, and every formula on this page handles that case unchanged.
Divide the later term by the first, then take the root corresponding to the number of steps between them — the difference in their positions.
Yes. Each repeated block is the previous one divided by a power of ten, so the infinite sum formula converts any repeating decimal into an exact fraction.
This page explains a mathematical calculation for educational purposes only. It is not financial or investment advice, and projections built on a fitted ratio should never be used as the sole basis for a decision.
