What is a geometric sequence in simple terms?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same constant number, called the common ratio. For example, 2, 6, 18, 54, ... is geometric because each term is 3 times the one before it.

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

To find the common ratio, divide any term (after the first) by the term right before it. If the sequence is 5, 15, 45, 135, ... then 15 ÷ 5 = 3 and 45 ÷ 15 = 3, so the common ratio is 3.

How do you write a formula for a geometric sequence?

There are two common formulas. An explicit formula gives any term directly, usually a_n = a_1 · r^(n-1), where a_1 is the first term and r is the common ratio. A recursive formula gives a starting term and a rule like a_n = r · a_(n-1) to get each new term from the previous one.

What is a geometric sequence used for in real life?

Geometric sequences are used to model repeated growth or decay, such as interest on savings, population growth, or repeated discounts. When something is multiplied by the same factor at regular steps, a geometric sequence or a related exponential function is often a good model.

What Is a Geometric Sequence? Definition and Examples

Geometric sequence explained for kids

Imagine you start with 2 stickers, and every day you get twice as many as the day before. Your stickers go 2, 4, 8, 16, 32. Each time, you multiply by 2. That list of numbers is a geometric sequence because the amount changes by the same multiplication each step!

Step-by-step worked examples

Example 1 (basic middle school level): Identify the common ratio and find a term

Consider the geometric sequence: 3, 6, 12, 24, 48, ...

1. Find the common ratio. Divide a term by the one before it: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The common ratio is 2.

2. Write an explicit rule for the sequence. The first term is 3. For a geometric sequence, an explicit rule for the nth term is often written as a_n = a₁ · rⁿ⁻¹, where a₁ is the first term and r is the common ratio. Here, a_n = 3 · 2ⁿ⁻¹.

3. Use the rule to find the 6th term. Plug in n = 6: a₆ = 3 · 2⁶⁻¹ = 3 · 2⁵ = 3 · 32 = 96. So the 6th term is 96.

Example 2 (more advanced high school level): Write recursive and explicit rules and model a situation

A small online video has 500 views on day 0. Each day, the number of views is multiplied by 1.5 (a 50% increase). The sequence of views by day is geometric.

1. Identify the first term and common ratio. Day 0 has 500 views, so a₀ = 500. Each day the views are multiplied by 1.5, so the common ratio r = 1.5.

2. Write a recursive rule. A recursive rule defines each term using the one before it. For this sequence, a₀ = 500, and for n ≥ 1, a_n = 1.5 · a_n−1.

3. Write an explicit rule. An explicit rule gives a direct formula for any term. Since the domain starts at n = 0, the rule is a_n = 500 · 1.5ⁿ.

4. Use the explicit rule to find the number of views on day 4. Plug in n = 4: a₄ = 500 · 1.5⁴. First compute 1.5² = 2.25. Then 1.5⁴ = 2.25 · 2.25 = 5.0625. Now multiply: 500 · 5.0625 = 2,531.25. So the model predicts about 2,531 views on day 4.

Going deeper with geometric sequences

In algebra, a geometric sequence is closely connected to an exponential function. If the terms of a geometric sequence are written as f(x) = a · b^x with x = 0, 1, 2, 3, ... then the sequence is a special case of an exponential function whose domain is a subset of the integers.

Your child will learn two main ways to describe a geometric sequence: an explicit formula (for example, a_n = a₁ · rⁿ⁻¹) and a recursive formula (for example, a₁ = starting value, a_n = r · a_n−1). Being able to move between these forms helps them solve problems more flexibly.

Later, geometric sequences lead into geometric series, where terms are added together. Students use formulas for the nth term and for the sum of a finite geometric sequence to solve real-world problems, such as total savings with repeated deposits or total distance traveled with repeated jumps.

Common mistakes with geometric sequences

Mistake 1: Treating a geometric sequence like an arithmetic sequence. A student might think the sequence 3, 6, 12, 24, ... has a common difference of 3 and try to add 3 each time, which would give 27 instead of the correct next term 48. Correction: geometric sequences multiply by a constant ratio, they do not add a constant difference.
Mistake 2: Using the wrong exponent in the explicit formula. For the sequence 5, 10, 20, 40, ... a student might write a_n = 5 · 2ⁿ and get a₁ = 10 instead of 5. Correction: when the first term is a₁, the standard explicit formula is a_n = a₁ · rⁿ⁻¹, so here a_n = 5 · 2ⁿ⁻¹.
Mistake 3: Misidentifying the common ratio when it is a fraction or negative. For the sequence 16, 8, 4, 2, ... a student might say the ratio is 8 because 16 − 8 = 8, then continue by subtracting 8 to get −6. Correction: the common ratio is found by division, 8 ÷ 16 = 1/2, so each term is half the previous one and the next term after 2 is 1.

How geometric sequences show up in everyday life

Geometric sequences appear in many situations where something grows or shrinks by the same percentage or factor each step. Examples include money in a savings account that earns the same interest rate, the height of a bouncing ball that loses the same fraction of its height each bounce, or the number of people who see a video when each viewer shares it with a fixed number of friends.

Related math ideas

Geometric sequences connect to several other algebra topics your child may study:

What is a geometric sequence?