Geometric Sequence Calculator Explained: From the Formula to the Worked Example

What a geometric sequence actually is

A geometric sequence is a list of numbers in which each term is the previous one multiplied by the same fixed value. That multiplier is the common ratio, written r. The starting value is the first term, written a₁. Once you know a₁ and r, every later term is determined: a₂ = a₁ × r, a₃ = a₁ × r², and so on. The Geometric Sequence Calculator takes a₁, r, and n and returns the full list of terms, the nth term aₙ, the partial sum Sₙ, and — when |r| < 1 — the infinite sum S∞ the series converges to.

The defining property is the constant ratio. The list 3, 6, 12, 24, 48 is geometric because each step multiplies by 2. The list 3, 7, 11, 15 is not geometric, because each step adds 4 instead. Sequences with a constant difference are arithmetic, not geometric, and they follow a different pair of formulas. The Arithmetic Sequence Calculator handles that case.

The three formulas you actually need

Every output the calculator returns reduces to three short formulas. The first is the closed form for the nth term, the second is the closed form for the partial sum, and the third is the limit of that partial sum as n grows without bound.

Nth term:     aₙ = a₁ × r^(n − 1)
Partial sum:  Sₙ = a₁ × (1 − rⁿ) / (1 − r),  r ≠ 1
Infinite sum: S∞ = a₁ / (1 − r),  |r| < 1

These three lines do the entire job. The partial-sum formula is what lets a 1,000-term sequence resolve instantly inside the calculator: nothing is summed term by term, the closed form Sₙ is evaluated directly from a₁, r, and n. The infinite sum is the limit of Sₙ as n grows, and only exists when the terms shrink fast enough — formally, when |r| is strictly less than 1.

These formulas appear in chapter 9 of OpenStax's Algebra and Trigonometry and in every secondary textbook that covers sequences. Geometric sequences are the simplest non-trivial example of exponential growth (or decay) sampled at integer times, which is why they sit at the foundation of compound interest, radioactive decay, population dynamics, and the geometric series that drives most of first-year calculus.

Where the formulas come from

The nth term is just counting multiplications by r

Start at a₁ and ask how many times you have multiplied by r by the time you reach the nth term. The first term has been multiplied by r zero times. The second, once. The third, twice. By induction, the nth term has been multiplied by r exactly (n − 1) times, so aₙ = a₁ × r^(n − 1). The off-by-one — (n − 1) instead of n — is identical to the arithmetic case and trips students up just as often. A quick sense-check: plug n = 1 and you should get a₁, not a₁ × r.

The partial-sum trick: multiply, subtract, divide

The derivation of Sₙ is one of the prettiest in elementary algebra. Write the sum out:

Sₙ      = a₁ + a₁r + a₁r² + … + a₁r^(n−1)

Now multiply both sides by r:

r × Sₙ  = a₁r + a₁r² + a₁r³ + … + a₁rⁿ

Subtract the second equation from the first. Every term in the middle cancels — a₁r cancels with a₁r, a₁r² with a₁r², all the way to a₁r^(n−1). What survives is the first term of the first line and the last term of the second line:

Sₙ − r Sₙ = a₁ − a₁rⁿ
Sₙ (1 − r) = a₁ (1 − rⁿ)
Sₙ = a₁ × (1 − rⁿ) / (1 − r)

The "multiply by r, then subtract" trick is the same idea Gauss used for the arithmetic series, but applied multiplicatively. The condition r ≠ 1 enters at the last step — you cannot divide by 1 − r when r = 1. The r = 1 case is handled separately: every term equals a₁, so Sₙ = n × a₁ by direct counting.

The infinite sum is what happens to rⁿ as n grows

Look at the closed-form Sₙ. The only piece that depends on n is rⁿ. If |r| < 1, then rⁿ shrinks toward zero as n grows, so Sₙ approaches a₁ × (1 − 0) / (1 − r) = a₁ / (1 − r). That limit is the infinite sum S∞, and it is finite. If |r| ≥ 1, rⁿ either stays the same size (r = ±1) or grows without bound (|r| > 1), and Sₙ has no finite limit. This is the whole content of the geometric series convergence test.

Worked example 1: a growing sequence

Take a₁ = 3 and r = 2 with n = 5. Plug those into the Geometric Sequence Calculator and the list is 3, 6, 12, 24, 48.

The fifth term is a₅ = 3 × 2⁴ = 3 × 16 = 48 — matching the last entry, as it should. The partial sum is S₅ = 3 × (1 − 2⁵) / (1 − 2) = 3 × (1 − 32) / (−1) = 3 × (−31) / (−1) = 93. Add the terms by hand to confirm: 3 + 6 + 12 + 24 + 48 = 93. Both calculations agree. Since |r| = 2 ≥ 1, this series diverges and there is no S∞ — the partial sums march off to infinity.

Worked example 2: a shrinking, converging series

Take a₁ = 1 and r = ½ with n = 6. The list is 1, 0.5, 0.25, 0.125, 0.0625, 0.03125. The sixth term is a₆ = 1 × (0.5)⁵ = 0.03125. The partial sum is S₆ = 1 × (1 − 0.5⁶) / (1 − 0.5) = 1 × (1 − 1/64) / 0.5 = (63/64) / 0.5 = 63/32 = 1.96875.

The interesting answer here is the infinite sum. Because |r| = 0.5 < 1, the series converges to S∞ = 1 / (1 − 0.5) = 1 / 0.5 = 2. No matter how many terms you add, you can never quite reach 2 — but you can get arbitrarily close. This is the classical Zeno-style geometric series at the root of most "Achilles and the tortoise" paradoxes: the infinite sum of halving distances is finite.

Worked example 3: an alternating sequence

Take a₁ = 5 and r = −2 with n = 6. The list is 5, −10, 20, −40, 80, −160. Every other term flips sign, because a negative ratio multiplies through alternately. The sixth term is a₆ = 5 × (−2)⁵ = 5 × (−32) = −160 — which agrees with the last entry.

The partial sum is S₆ = 5 × (1 − (−2)⁶) / (1 − (−2)) = 5 × (1 − 64) / 3 = 5 × (−63) / 3 = −105. Hand-add the terms to check: 5 − 10 + 20 − 40 + 80 − 160 = −105. Because |r| = 2 ≥ 1, the series still diverges, but the partial sums oscillate rather than running off in one direction. A negative ratio between −1 and 0 — say r = −½ — would converge, and the infinite sum would be S∞ = a₁ / (1 − r) = 5 / (1 − (−0.5)) = 5 / 1.5 ≈ 3.333.

Where geometric sequences appear in the real world

Compound interest and inflation

Money in a savings account that compounds at a fixed rate grows as a geometric sequence. A balance of 1,000 at 4% compounded annually generates 1,000, 1,040, 1,081.60, 1,124.86, … — a geometric sequence with a₁ = 1,040 and r = 1.04. Inflation works the same way in reverse: a fixed annual inflation rate shrinks real purchasing power geometrically. For percentage arithmetic on a single compounding period, the Percentage Calculator handles the underlying multiplications.

Radioactive decay and half-lives

A radioactive sample loses a fixed fraction of its mass each half-life, producing a geometric sequence with r = ½. After one half-life, half the original mass remains; after two half-lives, a quarter; after n half-lives, (½)ⁿ of the original. The same multiplicative-decay arithmetic applies to drug elimination from the bloodstream, capacitor discharge, and any first-order rate process sampled at equal time intervals.

Population growth at a fixed rate

A population that grows by a fixed percentage per generation — bacteria in unconstrained nutrient, an invasive species with no predator, a viral infection in its exponential phase — generates a geometric sequence of population sizes. The eventual carrying-capacity ceiling is what breaks the geometric model and forces logistic-growth replacements, but for the early phase the geometric sequence is an excellent fit.

The "0.999… = 1" identity

The recurring decimal 0.999… is the geometric series 9/10 + 9/100 + 9/1000 + … with a₁ = 9/10 and r = 1/10. The infinite sum is S∞ = (9/10) / (1 − 1/10) = (9/10) / (9/10) = 1. The same construction shows that any recurring decimal equals a rational number — 0.142857142857… is the geometric sum that comes out to 1/7. The infinite geometric series is the bridge between decimals and fractions.

Common mistakes

Treating r^(n − 1) as rⁿ

Same off-by-one as the arithmetic case, only multiplicative. Students see "the nth term" and write a₁ × rⁿ, which gives the (n + 1)th term. The sense-check is n = 1: the formula must return a₁. a₁ × r⁰ = a₁ × 1 = a₁, correct. a₁ × r¹ = a₁ × r, wrong. If the first term comes out wrong, you have the wrong exponent.

Trying to take the infinite sum when |r| ≥ 1

The infinite-sum formula a₁ / (1 − r) only applies when |r| is strictly less than 1. Plug r = 2 into it and you get a₁ / (−1), which is not the actual infinite sum (the series diverges, the answer is +∞). The closed form looks innocuous, but the convergence condition is essential. Check |r| < 1 before quoting S∞.

Dividing by zero when r = 1

The partial-sum formula has (1 − r) in the denominator. At r = 1 this is zero and the formula breaks. But the r = 1 sequence is the constant sequence a₁, a₁, a₁, … and its sum is trivially n × a₁. The calculator handles this case separately. Pen-and-paper users have to remember the special case manually.

Confusing geometric with exponential

Geometric sequences are exponential functions sampled at integer points. Continuous exponential growth (e^(kt)) and discrete geometric growth (a₁ × r^(n−1)) are the same underlying idea evaluated on different domains. In an applied problem, watch whether the rate is given as a per-period multiplier (geometric, use r) or as a continuously compounded rate (exponential, use e^k = r). Mixing the two is the most common applied-math error in finance and biology problems.

When the calculator is not enough

For one-off school problems the closed-form formulas and this calculator are the whole job. For richer questions you may need:

• Compound interest with explicit payment schedules — savings deposits, loan repayments, or mortgage amortisation are arithmetico-geometric, not purely geometric, because there is a fixed cash flow per period on top of the multiplicative growth. A dedicated finance calculator handles the combined formula.
• The geometric mean of a list of numbers is the nth root of their product. It is the natural average to use when the underlying process is multiplicative (returns, growth rates, ratios). The Geometric Mean Calculator handles arbitrary lists.
• Sequences that are neither arithmetic nor geometric — Fibonacci, factorials, prime gaps — have no closed-form term-by-term shortcut of the same shape. The Number Sequence Calculator covers the arithmetic and geometric cases under one interface; anything else needs a custom approach.

For a head-to-head comparison of the arithmetic and geometric closed forms, the Arithmetic Sequence Calculator is the right next stop — most students benefit from running the same a₁ and number of terms through both calculators with d = r to see the structural contrast.

Frequently asked questions

What is the difference between a geometric sequence and a geometric series?

A sequence is the list of terms: 3, 6, 12, 24, 48. A series is what you get when you add them up: 3 + 6 + 12 + 24 + 48 = 93. The Geometric Sequence Calculator returns both — the listed terms, the partial sum Sₙ, and when applicable the infinite sum S∞. "Geometric progression" (GP) is the British school-textbook synonym for geometric sequence.

When does the infinite sum exist?

Only when |r| < 1. In that case rⁿ shrinks to zero as n grows, and Sₙ approaches the finite limit a₁ / (1 − r). At |r| ≥ 1 the terms either stay the same size (|r| = 1) or grow (|r| > 1), and Sₙ has no finite limit. The boundary cases r = 1 and r = −1 both produce divergent series even though the terms themselves don't grow.

Can the common ratio be negative, fractional, or irrational?

Yes to all three. A negative ratio produces an alternating sequence (5, −10, 20, …). A fractional ratio between 0 and 1 produces a shrinking sequence that converges (1, ½, ¼, …). An irrational ratio like √2 produces a perfectly valid sequence whose terms grow without obvious pattern (1, √2, 2, 2√2, 4, …). The only forbidden ratio in normal use is r = 0, which would make every term after the first equal to zero — technically a geometric sequence by some definitions, but uninteresting.

How is the partial-sum formula derived?

Write Sₙ = a₁ + a₁r + … + a₁r^(n−1). Multiply both sides by r to get r Sₙ = a₁r + a₁r² + … + a₁rⁿ. Subtract: every middle term cancels and you are left with Sₙ − r Sₙ = a₁ − a₁rⁿ, so Sₙ (1 − r) = a₁ (1 − rⁿ). Divide by (1 − r): Sₙ = a₁ × (1 − rⁿ) / (1 − r). The r = 1 case has to be handled separately because the division would be by zero.

What does the "0.999… = 1" identity have to do with this?

0.999… is the infinite geometric series 0.9 + 0.09 + 0.009 + …, with a₁ = 0.9 and r = 0.1. Plug into the infinite-sum formula: S∞ = 0.9 / (1 − 0.1) = 0.9 / 0.9 = 1. That is the cleanest one-line proof that 0.999… is exactly 1, not just very close to it. The same trick converts any recurring decimal into its rational form.

How does compound interest relate to a geometric sequence?

A balance compounded annually at rate r grows by the factor (1 + r) each year. After n years the balance is P × (1 + r)ⁿ, which is the (n + 1)th term of a geometric sequence with first term P and common ratio (1 + r). That is the compound-interest formula in geometric-sequence clothing. The connection is the reason banks talk about "compound growth" and mathematicians talk about "geometric growth" — it's the same arithmetic.

Is there a limit on the number of terms?

The calculator caps n at 1,000 to keep the listed output readable in a browser. The closed-form aₙ and Sₙ formulas work for any positive integer n, so for sequences longer than 1,000 terms you can still compute the nth term and partial sum by hand without listing every value. That is the whole point of the closed form.

What's the difference between geometric and exponential growth?

They are the discrete and continuous versions of the same idea. A geometric sequence samples an exponential function at integer times: aₙ = a₁ × r^(n−1) is the same shape as f(t) = A × e^(kt) with r = e^k. In applied problems geometric models per-period rates (monthly returns, annual decay) and exponential models continuous rates (continuous compounding, instantaneous decay). They are convertible — a 5% per-period rate corresponds to a continuous rate of ln(1.05) ≈ 4.879%.