Geometric Sequences — Mathematics Knowledge System

Overview

A geometric sequence is a sequence in which each term is obtained from the previous one by multiplying by a constant ratio r. Because growth is multiplicative rather than additive, geometric sequences correspond to exponential functions of the index n. When |r| < 1, the corresponding infinite geometric series converges to a finite sum — a surprising and useful fact behind everything from repeating decimals to the economics of stimulus spending.

Intuition

Where an arithmetic sequence adds the same amount each step, a geometric sequence multiplies by the same factor each step — like money compounding, a population doubling, or a ball bouncing to a fixed fraction of its previous height each time. If that fraction is less than 1, each new term shrinks, and remarkably, infinitely many shrinking terms can still add up to a finite total, because each term is geometrically dwarfed by the sum of all the ones before it.

Interactive Graph

Formal Definition

Definition

A geometric sequence has a constant common ratio r between consecutive terms:

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

a₁ is the first term, r is the common ratio (r≠0)

Explicit formula

a_n = r\cdot a_{n-1}

Recursive formula

Derivation

Deriving the finite geometric sum formula by multiplying Sₙ by r and subtracting:

S_n = a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1}

Write out the sum

r S_n = a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1} + a_1 r^n

Multiply every term by r

S_n - rS_n = a_1 - a_1 r^n

Subtract; all the middle terms cancel, telescoping

S_n(1-r) = a_1(1-r^n) \implies S_n = a_1\frac{1-r^n}{1-r}

Solve for Sₙ (valid since r≠1)

Properties

Common ratio

r = \frac{a_n}{a_{n-1}} \text{ (constant for all } n\text{)}

Finite geometric series sum

S_n = a_1\cdot\frac{1-r^n}{1-r}, \quad r \neq 1

Infinite geometric series sum

S = \frac{a_1}{1-r}, \quad |r| < 1

Divergence for |r|≥1

\text{The infinite series has no finite sum when } |r| \geq 1

Applications

Compound interest and the value of an ordinary annuity are both geometric series; loan amortization schedules rely on the same closed-form sum.

Worked Examples

Identify a₁=3 and r=2, then apply the explicit formula.
$a_6 = 3\cdot 2^{6-1} = 3\cdot 32 = 96$

Answer: a₆ = 96

Practice Problems

Difficulty 3/10

Find the common ratio and the 5th term of the sequence 2, -6, 18, -54, ...

Difficulty 5/10

Express the repeating decimal 0.7777... as a fraction using an infinite geometric series.

Common Mistakes

Common Mistake

Applying the infinite sum formula S=a₁/(1-r) even when |r|≥1.

The infinite geometric series only converges to a finite sum when |r|<1. If |r|≥1, the series diverges (grows without bound or oscillates), and the formula S=a₁/(1-r) is meaningless there.

Common Mistake

Mixing up the exponent in aₙ=a₁rⁿ⁻¹, e.g. using rⁿ instead of rⁿ⁻¹.

The first term requires zero multiplications by r (a₁ = a₁·r⁰), so the correct exponent is (n-1). Check: a₁ = a₁r^(1-1) = a₁r⁰ = a₁ ✓.

Summary

A geometric sequence has a constant common ratio r: aₙ = a₁rⁿ⁻¹.
The finite sum is Sₙ = a₁(1-rⁿ)/(1-r), derived by the multiply-and-subtract telescoping trick.
The infinite sum S = a₁/(1-r) exists only when |r| < 1; otherwise the series diverges.
Geometric sequences model multiplicative (exponential) change: compound interest, bouncing-ball heights, recursive algorithm subproblem sizes.

References

BookSullivan, M. Algebra and Trigonometry, 10th ed. Ch. 12 — Sequences, Induction, and Probability.
WebsiteWikipedia — Geometric progression