Geometric Sequence Calculator – calculatorpeak.com

About the Geometric Sequence Calculator

A geometric sequence multiplies by the same amount every step. 2, 6, 18, 54, 162 — always three times the last. That fixed multiplier is the common ratio, and the difference between multiplying and adding turns out to be enormous.

Starting from 3, an arithmetic sequence adding 3 each step reaches 30 by its tenth term. A geometric one multiplying by 3 reaches 59,049. Multiplying compounds, and compounding is why interest, populations, viral spread and Moore's law all behave in ways that surprise people who expect steady growth.

This calculator finds any term, sums any number of them, recovers the ratio from two terms, and — where the series converges — sums infinitely many.

How to Use the Geometric Sequence Calculator

From the first term and ratio. Lists the terms and their sum.

From two known terms. Recovers the rule, and warns you when two answers fit.

Sum to infinity. Only meaningful when the ratio is between −1 and 1. If it is not, the calculator says so instead of returning a number.

The ratio can be negative (the terms alternate in sign) or fractional (they shrink).

The Formulas

  The nth term:   aₙ = a₁ · r^(n−1)

  The sum:        Sₙ = a₁(1 − rⁿ) / (1 − r)        for r ≠ 1

  To infinity:    S∞ = a₁ / (1 − r)                 only when |r| < 1

The nth term uses n − 1 for the same reason the arithmetic one does: the first term has not been multiplied yet.

At r = 1 the sum formula divides by zero, so it needs its own case: every term is the same, and the sum is just n × a₁. This calculator handles it separately rather than returning an error or a NaN.

Everything Depends on the Ratio

| Ratio | Behaviour | Infinite sum? | |-------|-----------|---------------| | r > 1 | grows, accelerating | no — unbounded | | r = 1 | constant | no — grows without limit | | 0 < r < 1 | shrinks towards zero | yes | | r = 0 | everything after the first term is zero | trivially yes | | −1 < r < 0 | shrinks, alternating sign | yes | | r = −1 | flips between two values forever | no — never settles | | r < −1 | grows, alternating sign | no |

The dividing line is |r| < 1. Inside it the terms shrink fast enough that adding infinitely many gives a finite total. Outside it they do not, and no finite sum exists.

How an endless sum can be finite

Take a half, a quarter, an eighth, a sixteenth, and keep going forever:

  0.5 + 0.25 + 0.125 + 0.0625 + … = 1

Each term covers half of what is left. You get closer and closer to 1 and never pass it. After ten terms you are at 0.999, after twenty at 0.999999. The total does not "eventually reach" 1 — it is 1, in the sense that no smaller number is an upper bound.

This is what the formula S∞ = a₁/(1 − r) captures: 0.5 / (1 − 0.5) = 1.

This calculator also shows how many terms get within 0.1% of the limit, which makes "converges" a concrete claim rather than an abstract one.

And why the calculator refuses when it does not converge

At r = 2 the terms are 1, 2, 4, 8, 16… The sum passes any number you care to name. There is no limit, and the formula would give a₁/(1 − 2) = −a₁ — a negative answer for a sum of positive terms, which is nonsense.

That absurd result is a well-known trap: it comes from applying a formula outside the conditions it was derived under. The calculator checks |r| < 1 first and explains rather than printing it.

Step-by-Step Example

Finding a term. Starting at 2, tripling each step. The 8th term?

  a₈ = 2 × 3⁷ = 2 × 2187 = 4374

And the sum of those eight terms:

  S₈ = 2(1 − 3⁸) / (1 − 3) = 2(1 − 6561) / (−2) = 6560

Note that the sum is barely larger than the last term alone. In a fast-growing geometric series the final term dominates everything before it — the first seven terms together come to only 2,186.

Recovering the ratio. The 2nd term is 6 and the 5th is 162.

  162 / 6 = 27, and that is r³
  r = ∛27 = 3
  a₁ = 6 / 3 = 2

The two-answer case. The 1st term is 2 and the 3rd is 18.

  18 / 2 = 9, and that is r²
  r = 3  or  r = −3

Both fit. 2, 6, 18 and 2, −6, 18 each have a first term of 2 and a third term of 18. An even gap between the two positions hides the sign, so there are always two answers — the calculator names both.

An even gap with a negative quotient, by contrast, has no answer at all: r² can never be negative.

Sum to infinity. First term 1, ratio 0.5.

  |0.5| < 1, so it converges.
  S∞ = 1 / (1 − 0.5) = 1 / 0.5 = 2

Understanding Your Result

The result answers the question asked.

The rule is given as aₙ = a₁ × r^(n−1).

The terms list the sequence.

The sum gives the total of the terms shown.

The does it converge line is the important one. For a shrinking sequence it gives the infinite sum; otherwise it explains why there is not one.

When Should You Use This Calculator?

Compound interest. Each year's balance is the last multiplied by (1 + rate) — a geometric sequence exactly.

Population and bacterial growth. Doubling at fixed intervals.

Radioactive decay and half-lives. A ratio of 0.5 per half-life.

Depreciation, reducing-balance. An asset losing a fixed percentage each year rather than a fixed amount.

Loan and annuity mathematics. The present value of a stream of payments is a geometric series, and the closed form is why mortgage formulas look the way they do.

Bouncing balls and damped motion. Each bounce reaches a fixed fraction of the last height, and the total distance is an infinite geometric sum.

Fractals and repeated subdivision. Each iteration scales by a fixed ratio.

Music. Equal temperament multiplies frequency by the twelfth root of two per semitone — a geometric sequence, which is why octaves sound evenly spaced.

Common Mistakes

Confusing the ratio with the difference. Divide consecutive terms to find r; do not subtract. If dividing gives a constant, the sequence is geometric.

Using n instead of n − 1. Gives a term one place too far along.

Applying the sum formula at r = 1. It divides by zero. That case is n × a₁.

Using S∞ when |r| ≥ 1. Produces an answer that is often negative and always meaningless.

Missing the second ratio. An even gap between two known terms admits both +r and −r.

Expecting slow growth. A ratio just above 1 looks harmless and compounds relentlessly: 1.05 per step reaches double in 15 steps and ten times in 48.

Rounding the ratio. Small errors in r are raised to the nth power, so they magnify enormously. Keep the exact fraction where you can.

Assuming shrinking means reaching zero. The terms of a converging series approach zero but never arrive, and the sum approaches its limit the same way.