Geometric Sequence

Q: What do students usually get wrong about Geometric Sequence?

If $|r| 1$, terms grow without bound.

Calculus

definition

Also known as: geometric progression

Grade 9-12

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r. Models compound interest, population growth, radioactive decay.

This concept is covered in depth in our geometric sequences with common ratios, with worked examples, practice problems, and common mistakes.

Definition

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r.

💡 Intuition

Multiply by the same number each step — 2, 6, 18, 54, ... (multiply by 3). This is exponential growth.

🎯 Core Idea

Geometric sequences represent exponential growth (|r|>1) or decay (|r|<1) — their graph is an exponential curve.

Example

3, 6, 12, 24, 48, ... — common ratio r = 2; the nth term is 3 \cdot 2^{n-1}.

Formula

a_n = a_1 \cdot r^{n-1}

Notation

r = common ratio, a_1 = first term, S_n = a_1 \cdot \frac{1 - r^n}{1 - r} = sum of first n terms (r \neq 1).

🌟 Why It Matters

Models compound interest, population growth, radioactive decay.

💭 Hint When Stuck

Divide any term by the previous one to find r, then check that ratio stays constant throughout.

Formal View

A sequence (a_n) is geometric if \exists r \in \mathbb{R} \setminus \{0\} : \frac{a_{n+1}}{a_n} = r for all n \geq 1. General term: a_n = a_1 \cdot r^{n-1}. Partial sum: S_n = a_1 \cdot \frac{1 - r^n}{1 - r} for r \neq 1.

Related Concepts

Sequence Exponents Series Exponential Function

🚧 Common Stuck Point

If |r| < 1, terms shrink toward zero. If |r| > 1, terms grow without bound.

⚠️ Common Mistakes

Confusing the common ratio r with the common difference d: in a geometric sequence you multiply by r, so r = \frac{a_{n+1}}{a_n}, not a_{n+1} - a_n.
Using r^n instead of r^{n-1} in the formula: a_n = a_1 \cdot r^{n-1}, not a_1 \cdot r^n — the first term has exponent 0, not 1.
Forgetting that a negative ratio makes terms alternate in sign: a_1 = 2, r = -3 gives 2, -6, 18, -54, ... — the absolute values grow but signs alternate.

Go Deeper

Worked Examples

Step-by-step solved problems

Practice Problems

Test your understanding

Formula Explained

Notation, derivation, and common mistakes

a_n = a_1 \cdot r^{n-1}

Frequently Asked Questions

What is Geometric Sequence in Math?

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r.

Why is Geometric Sequence important?

Models compound interest, population growth, radioactive decay.

What do students usually get wrong about Geometric Sequence?

If |r| < 1, terms shrink toward zero. If |r| > 1, terms grow without bound.

What should I learn before Geometric Sequence?

Before studying Geometric Sequence, you should understand: sequence, exponents.

Prerequisites

sequence exponents

Next Steps

series exponential function

Cross-Subject Connections

exponential function — Geometric sequences are the discrete version of exponential functions scaling — Each term is a fixed scaling of the previous term percent change — Constant percent change produces a geometric sequence

How Geometric Sequence Connects to Other Ideas

To understand geometric sequence, you should first be comfortable with sequence and exponents. Once you have a solid grasp of geometric sequence, you can move on to series and exponential function.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →

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Visualization

Static

Visual representation of Geometric Sequence