How do you find the common ratio of a geometric sequence?

Divide any term by the term immediately before it: $r = \dfrac{a_{n+1}}{a_n}$. If this ratio is the same for every consecutive pair, you have a geometric sequence. The common ratio can be positive, negative, or a fraction.

When does an infinite geometric series converge?

An infinite geometric series converges only when the absolute value of the common ratio is less than 1, that is $|r| < 1$. In that case the sum equals $\dfrac{a_1}{1 - r}$. If $|r| \geq 1$, the terms do not approach zero, so the series diverges and has no finite sum.

Can the common ratio be negative?

Yes. A negative common ratio causes the terms to alternate in sign. For example, 4, −8, 16, −32, … has $r = -2$. The finite-sum formula still works. For infinite sums, you need $|r| < 1$, so $r$ must be between −1 and 1 (exclusive).

Geometric Sequences and Series — Definition, Formula & Examples

Q: Can the common ratio be negative?

Yes. A negative common ratio causes the terms to alternate in sign. For example, 4, −8, 16, −32, … has $r = -2$. The finite-sum formula still works. For infinite sums, you need $|r| < 1$, so $r$ must be between −1 and 1 (exclusive).

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio. A geometric series is the sum of the terms in a geometric sequence.

A geometric sequence $\{a_n\}$ is defined by $a_n = a_1 \cdot r^{n-1}$ , where $a_1$ is the first term, $r$ is the common ratio ( $r \neq 0$ ), and $n$ is the term number. The corresponding geometric series is the sum $S_n = \sum_{k=1}^{n} a_1 \cdot r^{k-1}$ . When $|r| < 1$ , the infinite geometric series converges to $S = \dfrac{a_1}{1 - r}$ ; otherwise, the infinite series diverges.

Key Formula

a_n = a_1 \cdot r^{n-1} \qquad S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \qquad S_{\infty} = \frac{a_1}{1 - r} \;\;(|r| < 1)

Where:

$a_n$ = The nth term of the sequence
$a_1$ = The first term of the sequence
$r$ = The common ratio (each term divided by the previous term)
$n$ = The term number or the number of terms being summed
$S_n$ = The sum of the first n terms (finite geometric series)
$S_{\infty}$ = The sum of all terms in an infinite geometric series (exists only when |r| < 1)

How It Works

To identify a geometric sequence, divide any term by the previous term — if you always get the same value, that constant is the common ratio

r

. Once you know

a_1

and

r

, you can jump to any term using

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

without listing every term in between. To add up a finite number of terms, use the partial-sum formula

S_n = a_1 \cdot \dfrac{1 - r^n}{1 - r}

. If

|r| < 1

, the terms shrink toward zero, so the infinite sum settles on a finite value:

S = \dfrac{a_1}{1 - r}

. If

|r| \geq 1

, the terms do not shrink, and the infinite series has no finite sum — it diverges. Recognizing whether

|r|

is less than, equal to, or greater than 1 is the key decision point for most problems.

Worked Example

Problem: Find the 8th term and the sum of the first 8 terms of the geometric sequence 3, 6, 12, 24, …

Step 1: Identify the first term and the common ratio. The first term is 3, and dividing any term by the one before it gives the ratio.

a_1 = 3, \quad r = \frac{6}{3} = 2

Step 2: Use the nth-term formula with n = 8.

a_8 = 3 \cdot 2^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384

Step 3: Apply the finite-sum formula with n = 8.

S_8 = 3 \cdot \frac{1 - 2^8}{1 - 2} = 3 \cdot \frac{1 - 256}{-1} = 3 \cdot 255 = 765

Answer: The 8th term is 384, and the sum of the first 8 terms is 765.

Another Example

This example involves an infinite sum with |r| < 1, whereas the first example computed a finite sum with r > 1. It shows how to check for convergence before using the infinite-sum formula.

Problem: Find the sum of the infinite geometric series 100 + 50 + 25 + 12.5 + …

Step 1: Identify the first term and common ratio.

a_1 = 100, \quad r = \frac{50}{100} = 0.5

Step 2: Check convergence: since |r| = 0.5 < 1, the infinite series converges.

Step 3: Apply the infinite-sum formula.

S_{\infty} = \frac{100}{1 - 0.5} = \frac{100}{0.5} = 200

Answer: The infinite series converges to 200.

Visualization

Why It Matters

Geometric sequences and series appear throughout precalculus, AP Calculus (as the basis for power series and convergence tests), and the SAT/ACT. In finance, compound interest and loan amortization follow geometric patterns — understanding these formulas lets you compute future account values or total interest paid. Engineers and scientists use geometric series to model signal decay, population dynamics, and fractal geometry.

Common Mistakes

Mistake: Using the infinite-sum formula when |r| ≥ 1

Correction: The formula

S = \dfrac{a_1}{1-r}

only works when

|r| < 1

. Always check this condition first; if it fails, the infinite series diverges.

Mistake: Writing the exponent as n instead of n − 1 in the nth-term formula

Correction: The correct formula is

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

. The exponent is one less than the term number because the first term already exists before any multiplications by r.

Mistake: Confusing the common ratio with the common difference

Correction: In a geometric sequence you multiply to get the next term (

r = a_{n+1}/a_n

). In an arithmetic sequence you add (

d = a_{n+1} - a_n

). Subtracting consecutive terms of a geometric sequence does not give a constant.

Check Your Understanding

The geometric sequence starts 5, 15, 45, … What is the 6th term?

Hint: First find r by dividing 15 by 5, then use

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

Answer:

a_6 = 5 \cdot 3^5 = 5 \cdot 243 = 1215

Does the infinite series

8 + 8 + 8 + \dots

converge or diverge?

Hint: Check the absolute value of the common ratio.

Answer: It diverges because

r = 1

, so

|r| \not< 1

Find the sum of the infinite series

27 + 9 + 3 + 1 + \dots

Hint: Identify

r = \frac{1}{3}

, confirm

|r| < 1

, then apply the infinite-sum formula.

Answer:

S = \dfrac{27}{1 - \frac{1}{3}} = \dfrac{27}{\frac{2}{3}} = 40.5

Related Terms

Common Ratio — The fixed multiplier that defines any geometric sequence
Arithmetic Sequence — Additive counterpart to the multiplicative geometric sequence
Arithmetic Series — Sum of an arithmetic sequence, compared with geometric series
Converge — Describes behavior when |r| < 1 for infinite series
Convergent Series — An infinite geometric series with |r| < 1 is convergent
Diverge — Describes behavior when |r| ≥ 1 for infinite series
Divergent Series — An infinite geometric series with |r| ≥ 1 is divergent
Convergent Sequence — A geometric sequence converges when |r| < 1
Divergent Sequence — A geometric sequence diverges when |r| ≥ 1
Bounded Sequence — Geometric sequences with |r| ≤ 1 are bounded