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. The nth term is the balance after n years, and the compound interest formula is the nth-term formula.

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.

First term b₁

Common ratio q

Term number n

Result

nth term4,374
Sum of the first n6,560
Infinite sumряд расходится
Formulab8 = 2 × 3^7
Opening terms2, 6, 18, 54, 162
Behaviourрастёт по модулю

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.

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. The nth term is the balance after n years, and the compound interest formula is the nth-term formula.

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.

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.

Useful links

All tools in Math & Statistics

Geometric Sequence Calculator

Result

Related tools

Arithmetic Sequence Calculator

Compound Interest Calculator: Future Value and Growth

Exponent Calculator

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