How to Solve Geometric Sequences? (+FREE Worksheet!)

Geometric sequences are number patterns where each term is found by multiplying the previous term by the same fixed number, called the common ratio. They appear in exponential growth and decay, financial calculations, and many real-world models. Understanding geometric sequences gives you a powerful tool for analyzing patterns and predicting future values in Algebra 1 and beyond.

Try it

Sequence Calculator

Open the full calculator →

Type the problems from this lesson and check each step.

Tutor-style math help

Solve Geometric Sequences: what to notice and how to work it

Series skill

Sequences list terms; series add terms. The first question is whether the pattern adds the same amount or multiplies by the same factor.

What to notice first

Identify the term number. Many mistakes happen because \(a_1\), \(a_n\), and the number of terms get mixed up.

Common student mistake

Do not use an arithmetic formula on a geometric pattern. Check differences and ratios before choosing a formula.

Key formulas and cues

\(a_n=a_1+(n-1)d\)

\(a_n=a_1r^{n-1}\)

\(S_n=\frac{n}{2}(a_1+a_n)\)

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

A reliable path

Compare termsLook for a common difference or common ratio.
Choose term or sumDecide whether the question asks for one term or a total.
Track nMake sure n is the position or number of terms the question uses.

Worked examples

Arithmetic sequence

Example: 5, 9, 13, 17..

Each term adds 4.
The common difference is 4.
Add 4 to continue.

Answer: \(21\)

Geometric sequence

Example: 3, 6, 12, 24..

Each term multiplies by 2.
The common ratio is 2.
Multiply 24 by 2.

Answer: \(48\)

Open pop-up practice

Try one before moving on

Try: Find the next term: 10, 7, 4, 1..

Answer: \(-2\). The pattern subtracts 3.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

What Is a Geometric Sequence?

A geometric sequence is a list of numbers in which each term after the first is obtained by multiplying the previous term by a constant called the common ratio (\(\color{blue}{r}\)). If \(\color{blue}{r > 1}\), the sequence grows; if \(\color{blue}{0 < r < 1}\), the sequence shrinks; if \(\color{blue}{r < 0}\), the terms alternate in sign.

Grade 8 Math for Students The Ultimate Step by Step Guide to Preparing for the Grade 8 Math Test

Rated 4.34 out of 5 based on 61 customer ratings

Satisfied 52 Students

Example: \(\color{blue}{3, 6, 12, 24, 48, \ldots}\) The common ratio is \(\color{blue}{r = 2}\).

Key Formulas for Geometric Sequences

Finding the Common Ratio

Divide any term by the term before it:

\(\color{blue}{r = \frac{a_{n}}{a_{n-1}}}\)

Example: in \(\color{blue}{81, 27, 9, 3, \ldots}\) the ratio is \(\color{blue}{r = \frac{27}{81} = \frac{1}{3}}\).

Explicit (nth-Term) Formula

The nth term of a geometric sequence is:

a_n = a₁ · r^{\(\color{blue}{n-1}\)}

where \(\color{blue}{a_{1}}\) is the first term and \(\color{blue}{r}\) is the common ratio.

Recursive Formula

Each term is the previous term multiplied by \(\color{blue}{r}\):

\(\color{blue}{a_{n} = a_{n-1} \cdot r}\), with \(\color{blue}{a_{1}}\) given.

Step-by-Step Summary

Identify the first term \(\color{blue}{a_{1}}\).
Find the common ratio: \(\color{blue}{r = \frac{a_{2}}{a_{1}}}\) (verify with other consecutive pairs).
Write the explicit formula: \(\color{blue}{a_{n} = a_{1} \cdot r^{n-1}}\).
Substitute \(\color{blue}{n}\) to find any desired term.

Watch: Introduction to Geometric Sequences (Video Lesson)

Khan Academy introduces geometric sequences, common ratios, and the intuition behind the formula:

Geometric Sequences – Worked Examples

Example 1: Find the 5th term of the sequence \(\color{blue}{3, 6, 12, 24, \ldots}\)

\(\color{blue}{a_{1} = 3}\), \(\color{blue}{r = 2}\).
\(\color{blue}{a_{5} = 3 \cdot 2^{4} = 3 \cdot 16 = 48}\)

Example 2: Find the 5th term of \(\color{blue}{81, 27, 9, 3, \ldots}\)

\(\color{blue}{a_{1} = 81}\), \(\color{blue}{r = \frac{1}{3}}\).
\(\color{blue}{a_{5} = 81 \cdot (\frac{1}{3})^{4} = 81 \cdot \frac{1}{81} = 1}\)

Example 3: Find the 5th term of \(\color{blue}{2, -6, 18, -54, \ldots}\)

\(\color{blue}{a_{1} = 2}\), \(\color{blue}{r = -3}\).
\(\color{blue}{a_{5} = 2 \cdot (-3)^{4} = 2 \cdot 81 = 162}\)

Example 4: Find the common ratio when \(\color{blue}{a_{1} = 5}\) and \(\color{blue}{a_{4} = 40}\).

\(\color{blue}{40 = 5 \cdot r^{3} \Rightarrow r^{3} = 8 \Rightarrow r = 2}\)

More Practice: Using Explicit Formulas of Geometric Sequences (Video)

Khan Academy demonstrates how to apply the explicit formula to find specific terms and work backwards to find the common ratio:

Exercises for Geometric Sequences

Find the common ratio: \(\color{blue}{5, 15, 45, 135, \ldots}\)
Find the 6th term: \(\color{blue}{2, 6, 18, 54, \ldots}\)
Find the 4th term: \(\color{blue}{256, 64, 16, \ldots}\)
Write the explicit formula for: \(\color{blue}{4, 8, 16, 32, \ldots}\)
Find \(\color{blue}{a_{7}}\) when \(\color{blue}{a_{1} = 1}\) and \(\color{blue}{r = -2}\).

Answers

\(\color{blue}{r = 3}\)
\(\color{blue}{a_{6} = 2 \cdot 3^{5} = 486}\)
\(\color{blue}{a_{4} = 256 \cdot (\frac{1}{4})^{3} = 4}\)
\(\color{blue}{a_{n} = 4 \cdot 2^{n-1}}\)
\(\color{blue}{a_{7} = 1 \cdot (-2)^{6} = 64}\)

Geometry for Beginners The Ultimate Step by Step Guide to Acing Geometry

Frequently Asked Questions

What is a geometric sequence in math?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed number, called the common ratio. For example, 3, 6, 12, 24, … multiplies by 2 each time, so the next term is 48. If the ratio is between 0 and 1, like in 80, 40, 20, 10, …, the terms shrink instead of grow.

What is the geometric sequence formula?

The explicit formula is aₙ = a₁ × rⁿ⁻¹, where a₁ is the first term, r is the common ratio, and n is the term number. For the sequence 5, 15, 45, …, a₁ = 5 and r = 3, so the 6th term is 5 × 3⁵ = 5 × 243 = 1,215. The recursive form is aₙ = aₙ₋₁ × r.

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

Divide any term by the term right before it: r = a₂ ÷ a₁. In 2, −6, 18, −54, …, the ratio is −6 ÷ 2 = −3, and checking another pair (18 ÷ −6 = −3) confirms it. If the quotients between neighboring terms are not all the same, the sequence is not geometric. A negative ratio makes the signs alternate.

How do you find the sum of a geometric sequence?

Use Sₙ = a₁(1 − rⁿ) ÷ (1 − r), for any ratio r ≠ 1. For the first 5 terms of 4, 8, 16, 32, 64, a₁ = 4 and r = 2: S₅ = 4(1 − 32) ÷ (1 − 2) = 4(−31) ÷ (−1) = 124. Adding the terms directly, 4 + 8 + 16 + 32 + 64 = 124, confirms the answer.

Are geometric sequences exponential?

Yes. A geometric sequence is an exponential function whose inputs are limited to whole-number term positions. The formula aₙ = a₁ × rⁿ⁻¹ has the variable in the exponent, just like y = a × bˣ. For example, 2, 6, 18, 54, … matches the function f(n) = 2 × 3ⁿ⁻¹. Graphed, the points curve upward when r > 1 rather than lying on a straight line.

How do you find a geometric sequence with two terms?

Write both terms with the formula and divide to remove a₁. Suppose a₂ = 6 and a₅ = 162. Then a₁r⁴ ÷ a₁r = r³ = 162 ÷ 6 = 27, so r = 3. Next, a₁ = 6 ÷ 3 = 2, giving the sequence 2, 6, 18, 54, 162, … and the formula aₙ = 2 × 3ⁿ⁻¹.

Related to This Article

Geometric Sequences

What people say about "How to Solve Geometric Sequences"?

No one replied yet.

How to Solve Geometric Sequences? (+FREE Worksheet!)