Geometric Sequence Calculator

Background.

Put one grain of rice on the first square of a chessboard, two on the second, four on the third, and keep doubling. The 64th square alone needs 2⁶³ grains, about 9.223 × 10¹⁸, and that single square holds one grain more than the other 63 squares put together. Multiplying by the same number at every step is the whole definition of a geometric sequence, and that runaway arithmetic is why compound interest and the early days of an epidemic look slow for a while and then suddenly enormous.

Run the same rule with a factor below 1 and it works in reverse. A drug concentration that halves with every half-life, a ball that rebounds to 60% of the height it fell from, a signal that loses the same fraction at each stage of a cable: every step is a fixed fraction of the one before, the terms shrink toward zero, and the total of all of them, even infinitely many, settles on a finite number. In the idealised model the ball never quite stops bouncing, yet dropped from 2 metres with that 60% rebound it travels a total of exactly 8 metres.

Four quantities pin a geometric sequence down: the first term a₁, the common ratio r, a position n and the value aₙ found there. Know three and the fourth follows, so the calculator solves in all four directions. You can find a later term, back out the growth factor between two known values, recover a starting value, or locate where a given value falls in the run. Whichever you choose, it also reports the sum of the first n terms, the explicit rule with your numbers in it, a preview of the terms, and a sentence on whether the infinite series converges.

Positions are counted from 1, so the eighth term sits seven multiplications after the first. Some references, the Encyclopedia of Mathematics among them, count from 0 and call the starting value a₀. The sequences are identical; only the labels shift by one, which matters when you copy n out of a textbook problem.

What is geometric sequence calculator?

A geometric sequence, also called a geometric progression, is a list of numbers in which every term after the first equals the previous term multiplied by the same non-zero constant, the common ratio r. Dividing any term by the one before it therefore always gives the same answer, and that is the quickest test of whether data are geometric: 6 ÷ 2, 18 ÷ 6 and 54 ÷ 18 all equal 3. When the terms are positive there is a second test: each one is the geometric mean of its two neighbours, as in 18² = 324 = 6 × 54.

The ratio sets the shape. Above 1 the terms grow; between 0 and 1 they shrink toward zero without ever reaching it; exactly 1 gives a constant list; and a negative ratio flips the sign at every step, so the terms alternate between positive and negative while their size grows or shrinks according to |r|. The ratio cannot be 0, because one zero term makes every later term zero.

Adding the terms gives a geometric series. A finite series stops after n terms; an infinite one asks whether the running total approaches a limit, which happens only when |r| is below 1. Any exponential growth or decay sampled at equal intervals is a geometric sequence: a balance growing 5% a year has r = 1.05, and a radioactive sample measured once per half-life has r = 0.5.

How to use this calculator.

1. 01Pick the unknown under 'What do you want to find?': the nth term, the common ratio, the first term or the number of terms. Whatever sits in the field you are solving for is ignored.
2. 02Enter the first term a₁ and the common ratio r as plain numbers. Convert percentages first: growth of 5% per step is r = 1.05, a fall of 20% per step is r = 0.8, and halving is r = 0.5. Neither a₁ nor r may be 0.
3. 03Set the number of terms n, a whole number from 1 to 100,000, counting the first term as term 1. If your problem counts years or periods after an opening value, n is one more than that count.
4. 04When you solve for the ratio, the first term or the count, type the known later value into 'Value of the nth term (aₙ)'. Copy it at full precision: in count mode a value rounded to the cent can be rejected as not being a term at all.
5. 05Read the main result, then the sum and the infinite-series line. The preview shows the opening six terms and the last one, which is the fastest way to spot a mistyped ratio or a sign you did not expect.

The formula.

The nth term comes from counting multiplications. Getting from term 1 to term n takes n − 1 steps, each one multiplying by r, so aₙ = a₁ · r^(n−1). Writing a₁ · rⁿ instead is the classic off-by-one error, and it puts every term out by a factor of r.

The sum uses a short trick. Write Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n−1), multiply the whole line by r, and subtract one line from the other: every middle term cancels, leaving Sₙ − rSₙ = a₁ − a₁rⁿ, so Sₙ = a₁(1 − rⁿ)/(1 − r). That step divides by 1 − r, so r = 1 is handled separately: every term equals a₁ and the sum is simply n × a₁. A rearranged form, Sₙ = (r·aₙ − a₁)/(r − 1), uses the last term directly and is the easier one to check by hand.

Let n grow without limit and rⁿ fades to zero only when |r| < 1; the partial sums then close in on S∞ = a₁/(1 − r). At |r| ≥ 1 nothing fades. With r = 1 the total climbs forever, with r = −1 it flips between a₁ and 0, and beyond that the swings widen, so the result is stated as divergent instead of being given a number.

The inverse solves rearrange the same equation. The first term is a₁ = aₙ / r^(n−1). The ratio is the (n − 1)th root of aₙ/a₁; with an even number of steps and a positive quotient both the positive and the negative root fit, and the positive one is reported, while a negative quotient over an even number of steps has no real root and is refused. The position comes from logarithms, n = 1 + ln|aₙ/a₁| ⁄ ln|r|. Real data rarely give a whole number exactly, so the result is rounded to the nearest position, the term there is recomputed, and your value is accepted only if it agrees to about one part in a billion, a check that also catches a wrong sign when r is negative. A ratio of exactly 1 or −1 cannot locate a position, since every term has the same size.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), College Algebra 2e, §9.3 'Geometric Sequences' — 'Each term of a geometric sequence increases or decreases by a constant factor called the common ratio', and the explicit formula aₙ = a₁r^(n−1). Retrieved 2026-07-29; open access.
2. [2]OpenStax (Rice University), College Algebra 2e, §9.4 'Series and Their Notations' — the finite geometric series Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1, and the infinite sum S = a₁/(1 − r) under the stated condition −1 < r < 1. Retrieved 2026-07-29; open access.
3. [3]Encyclopedia of Mathematics (European Mathematical Society), 'Geometric progression' — independent check: 'each one of which is equal to the preceding one multiplied by a number q ≠ 0', Sₙ = a₀(1 − qⁿ)/(1 − q), and 'If |q| < 1, the sum Sₙ tends to the limit S = a₀/(1 − q)'. Agrees with OpenStax on the worked example. Retrieved 2026-07-29; open access.
4. [4]Wolfram MathWorld, 'Geometric Series' — the closed form and the convergence condition, used as a second cross-check on the infinite-sum branch. Retrieved 2026-07-29; open access.
5. [5]OpenStax (Rice University), College Algebra 2e, §9.2 'Arithmetic Sequences' — cited for the additive/multiplicative contrast drawn throughout this page. Retrieved 2026-07-29; open access.