Geometric Sequence Calculator

Q: When does an infinite sum exist?

When the ratio is less than one in magnitude. Then S = b₁ / (1 − q). For 1 + 1/2 + 1/4 + 1/8 + … the sum is exactly 2.

Q: How does this relate to compound interest?

Directly: a deposit at 10 percent a year is a sequence with ratio 1.1. If the first term is the opening balance, the balance after n years is the (n + 1)th term, b₁ · qⁿ — the compound interest formula is the nth-term formula with the index shifted by one.

Q: What is Zeno's paradox?

The argument that Achilles never catches the tortoise because infinitely many segments remain. Those segments form a convergent geometric series — a finite distance in a finite time.

Q: Can the ratio be negative?

Yes, and then the signs alternate. The series still converges when the magnitude is under one, and the sum formula is unchanged.

Q: What is the eighth term of 2, 6, 18?

4,374 — the first term multiplied by the ratio seven times, 2 × 3⁷. The first eight terms come to 6,560.

Written byOleh Soloninupdated 27 September 2026

What you know

First term b₁

Common ratio q

Term number n

Result

nth term4,374
Sum of the first n6,560
Sum by formulaS₈ = 2 × (3⁸ − 1) ÷ (3 − 1) = 6560
Infinite sumthe series diverges
Formulab₈ = 2 × 3⁷ = 4374
Opening terms2, 6, 18, 54, 162
Behaviourgrows in magnitude

Each term larger by the same factor

The nth term, the sum of the first n terms, and the sum of the infinite series where it converges.

A geometric sequence multiplies by a constant each step. Compound interest, bacterial growth and — less happily — a debt with capitalised interest all follow this pattern.

The interesting part is the infinite sum. When the ratio is less than one in magnitude, the terms shrink fast enough that an infinite series adds up to a finite number. That fact is what dissolves Zeno's paradox about Achilles and the tortoise.

How to use

Enter the first term and the ratio.
Enter which term you need.
Check whether the infinite sum exists: it does only when the ratio is below one in absolute value, |q| < 1.

When the series converges and when it runs away

With a first term of 2 and a ratio of 3, the eighth term is 4,374 and the first eight sum to 6,560. Growth like that is explosive: every term is three times the one before it.

A ratio smaller than one in absolute value changes everything: the series converges and an infinite sum exists. For a first term of 2 and a ratio of 0.5 that sum is exactly 4, and no number of further terms can pass it.

First term 2. Worked out with this calculator.
Ratio	Eighth term	Sum of eight	Infinite sum
3	4,374	6,560	diverges
2	256	510	diverges
0.5	0.015625	3.9844	4

What this calculator shows

The nth term and the sum of the first n terms
The infinite sum, when the series converges
The opening terms and how the sequence behaves

What to keep in mind

The ratio cannot be zero, or the sequence stops after the first term.
An infinite sum exists only when |q| < 1; otherwise the series diverges and the calculator says so.

FAQs

When does an infinite sum exist?

When the ratio is less than one in magnitude. Then S = b₁ / (1 − q). For 1 + 1/2 + 1/4 + 1/8 + … the sum is exactly 2.

How does this relate to compound interest?

Directly: a deposit at 10 percent a year is a sequence with ratio 1.1. If the first term is the opening balance, the balance after n years is the (n + 1)th term, b₁ · qⁿ — the compound interest formula is the nth-term formula with the index shifted by one.

What is Zeno's paradox?

The argument that Achilles never catches the tortoise because infinitely many segments remain. Those segments form a convergent geometric series — a finite distance in a finite time.

Can the ratio be negative?

Yes, and then the signs alternate. The series still converges when the magnitude is under one, and the sum formula is unchanged.

What is the eighth term of 2, 6, 18?

4,374 — the first term multiplied by the ratio seven times, 2 × 3⁷. The first eight terms come to 6,560.

Worked example

The sequence 2, 6, 18, …

Input: b₁ = 2, q = 3, n = 8

Output: b₈ = 4374, sum 6560

Note: The eighth term is already over four thousand — that is exponential growth. At a ratio of 0.5 the same sequence would converge to a sum of 4.

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All tools in Math & Statistics

Geometric Sequence Calculator

Result

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Arithmetic Sequence Calculator

Compound Interest Calculator

Exponent Calculator

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