What Is a Geometric Sequence?

What Is a Geometric Sequence? Definition & Meaning

A geometric sequence (also called a geometric progression) is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, the sequence 2, 6, 18, 54, ... is geometric because you multiply by 3 each time. The first term is 2 (a₁ = 2) and the common ratio is 3 (r = 3).

Geometric sequences appear everywhere in math and real life — from calculating bank interest to understanding how populations grow or shrink. They are one of the most useful patterns you can learn.

Where Geometric Sequences Come From

The idea of geometric sequences has been around for thousands of years. Ancient mathematicians like Euclid (around 300 BCE) studied them as part of number theory. Later, mathematicians used them to solve problems in finance and science. The term "geometric progression" comes from the fact that each term is a constant multiple of the one before it — much like the dimensions of a geometric shape that scales uniformly.

Today, we use the same formulas that were developed centuries ago. The nth term formula aₙ = a₁ × r^(n-1) helps you find any term in the sequence without listing all the previous ones. And the sum formula Sₙ = a₁ × (1 - rⁿ) / (1 - r) lets you add up a certain number of terms quickly. For a deeper dive into these formulas, check out our Geometric Sequence Formulas page.

Why Geometric Sequences Matter

Geometric sequences describe situations where something grows or decays by a constant percentage each step. This is different from arithmetic sequences, which add or subtract a constant amount. Because multiplication can make numbers grow very fast or shrink very fast, geometric sequences model many real-world phenomena:

• Compound interest: Your savings account grows by a fixed percentage each year. The amount after each year is a geometric sequence. See our Geometric Sequences in Finance page for more.
• Population growth: If a population increases by 5% each year, the number of people forms a geometric sequence.
• Radioactive decay: The amount of a radioactive substance decreases by a fixed fraction each half-life.
• Technology: Moore's Law (transistors doubling every ~2 years) is a geometric sequence.

Understanding geometric sequences helps you make predictions: How much money will I have in 10 years? How many people will live in my city? Will a virus spread exponentially? The How to Calculate Geometric Sequences guide walks you through the steps.

How Geometric Sequences Are Used: A Worked Example

Let's look at a simple example. Suppose you start a business and your monthly profit starts at $3,000 and doubles every month (common ratio r = 2). Your profits form a geometric sequence with a₁ = 3000 and r = 2.

Question: What is your profit in the 5th month? What is your total profit over the first 5 months?

Step 1: Find the 5th term (a₅).

Use the nth term formula: aₙ = a₁ × r^(n-1). Plug in n = 5, a₁ = 3000, r = 2.

a₅ = 3000 × 2^(5-1) = 3000 × 2^4 = 3000 × 16 = 48,000

So in month 5, your profit is $48,000.

Step 2: Find the sum of the first 5 months (S₅).

Use the sum formula: Sₙ = a₁ × (1 - rⁿ) / (1 - r) when r ≠ 1. Here n = 5.

S₅ = 3000 × (1 - 2^5) / (1 - 2) = 3000 × (1 - 32) / (-1) = 3000 × (-31) / (-1) = 3000 × 31 = 93,000

Total profit over the first 5 months is $93,000.

Notice how fast the numbers grow — that's the power of a geometric sequence with r > 1. On the other hand, if |r| < 1 (like 0.5), the terms get smaller and approach zero. You can learn about different ranges on our Geometric Sequence Ranges page.

Common Misconceptions About Geometric Sequences

Many students confuse geometric sequences with arithmetic sequences. Remember: arithmetic adds a constant difference; geometric multiplies by a constant ratio.

Misconception 1: The common ratio must be a whole number. No! It can be any non-zero number, including fractions, decimals, and negative numbers. For instance, 16, 8, 4, 2, ... has r = 0.5 (a fraction). And 5, -10, 20, -40, ... has r = -2 (negative). When r is negative, the terms alternate signs.

Misconception 2: The sequence always grows. If |r| > 1, it grows (if positive) or grows in magnitude (if negative). But if |r| < 1, the terms get smaller and approach zero. If r = 1, all terms are equal to the first term. If r = -1, the sequence bounces between two values.

Misconception 3: The sum to infinity always exists. The sum to infinity formula S∞ = a₁ / (1 - r) only works when |r| < 1. If |r| ≥ 1, the sum diverges (goes to infinity or doesn't settle). For example, 1 + 2 + 4 + 8 + ... has no finite sum.

Misconception 4: You can't have a geometric sequence with zero terms. Actually, a geometric sequence can have zero if the first term is zero (then all terms are zero) — but usually we avoid zero because the common ratio would be undefined (0/0).

For more common questions, see our Geometric Sequence FAQs.