# Geometric Sequence – Pattern, Formula, and Explanation

Geometric sequences are a series of numbers that share a common ratio. We cab observe these in population growth, interest rates, and even in physics! This is why we understand what geometric sequences are.

Geometric sequences are sequences of numbers where two consecutive terms of the sequence will always share a common ratio.

We’ll learn how to identify geometric sequences in this article. We’ll also learn how to apply the geometric sequence’s formulas for finding the next terms and the sum of the sequence.

We’ll also learn how to identify geometric sequences from word problems and apply what we’ve learned to solve and address these problems. So, let’s begin by understanding the definition and conditions of geometric sequences.

## What is a geometric sequence?

Geometric sequences are sequences where the term of the sequence can be determined by multiplying the previous term with a fixed factor we call the common ratio.

The sequence above shows a geometric sequence where we multiply the previous term by $2$ to find the next term.

That’s why we have the following terms:

\begin{aligned} 1 \times 2 &= 2\\2 \times 2 &= 4\\4 \times 2 &= 8\\8 \times 2 &= 16\\.\\.\\. \end{aligned}

This shows that this sequence has a common ratio of $2$.

Let’s look at another way of understanding geometric sequences and common ratios by the figure shown below.

This figure is a visual representation of terms from a geometric sequence with a common ratio of $\dfrac{1}{2}$. We can find the smaller square dimensions by taking half of the length of the previous dimensions.

\begin{aligned} 1 \times \color{green}\dfrac{1}{2} &= \dfrac{1}{2}\\\dfrac{1}{2} \times \color{green}\color{green}\dfrac{1}{2} &= \dfrac{1}{4}\\\dfrac{1}{4} \times \color{green}\color{green}\dfrac{1}{2} &= \dfrac{1}{8}\\.\\.\\. \end{aligned}

This means that when you’re having a memory lapse and need a quick refresher of what a geometric sequence is – it helps to remember how it’s like when we fold papers into a half fold over and over again.

Geometric sequence definition

Now that we have an idea of what a geometric sequence is, we should define it formally so that we have its general form and we can use it to work on all types of geometric sequences.

Let’s say the first term of the sequence is $a$ and the sequence has a common ratio of $r$, and the second term can be determined by multiplying $a$ by $r$. This process continues throughout the entire process.

Keep in mind as well that $r$ can either be an integer or a rational factor.

• When $|r| < 1$, the next term becomes smaller (disregarding the sign of the term).
• When $|r| > 1$, the next term becomes bigger (disregarding the sign of the term).

Here are four more examples of geometric sequences. We’ll observe what happens when $a$ is positive and when $r$ is a whole number or a rational one.

 $\boldsymbol{a}$ $\boldsymbol{r}$ Geometric Sequence Positive, $a= 3$ $|r| > 1$, $r = 2$ $3, 6, 12, 24, …$ Positive, $a= 9$ $|r| < 1$, $r = \dfrac{1}{3}$ $9, 3, 1, \dfrac{1}{3}, …$ Negative, $a= -3$ $|r| > 1$, $r = 3$ $-3, -9, -27, -81, …$ Negative, $a= -8$ $|r| < 1$, $r = \dfrac{1}{2}$ $-8, -4, -2, -1, …$

These examples can help us better understand how the signs of the first term and the value of the common ratio affect the next terms. From this, we can also see that the terms of a geometric sequence drastically increase or decrease because we’re working with the common ratio’s powers, $r$.

## How to solve geometric sequences?

There are different approaches to finding the unknown elements of a geometric sequence. The most important step is finding the common ratio shared by the sequence since, in most formulas, $r$ is essential.

We’ll slowly dive right into these different formulas and understand when they are most useful in the next sections. Why don’t we begin with learning how we can find $a_n$.

### Geometric sequence formula

In this section, we’ll learn how to find the nth term, $a_n$, of a geometric sequence by expressing $a_n$ using the explicit and recursive rules.

Recursive Rule

We can find any term from the sequence using the recursive rule. For this, we’ll need the term before $a_n$ and the common ratio to find the value of $a_n$.

\begin{aligned}a_n = a_{n – 1} \cdot r\end{aligned}

This makes sense since we need to multiply the previous term, $a_{n-1}$, by the common ratio, $r$, to find the value of the next term, $a_n$.

We can also use this rule to find an expression for $r$ in terms of $a_{n – 1}$ and $a_n$.

\begin{aligned}r = \dfrac{a_n}{a_{n – 1}}\end{aligned}

However, this form has its limitations, especially when we want to find, let’s say, the $40$th term of a sequence. This is also why we need to learn about the explicit rule.

Explicit Rule

Let’s begin by observing the general expressions for the terms of a geometric sequence.

\begin{aligned} a_1 &= a\\a_2 &= a \cdot r\\a_3 &= ar \cdot r\\&= ar^2\\a_4&= ar^2 \cdot r\\&= ar^3\\.\\.\\.\end{aligned}

Notice anything? To find the nth term, we multiply the first term by the ratio raised to the $(n – 1)$th term.

This means that if we continue the equations show above and include $a_n$, we have the explicit rule shown below.