Lesson Objectives
• Learn how to find the common ratio
• Learn how to find the nth term of a geometric sequence
• Learn how to evaluate a geometric series

What is a Geometric Sequence?

In this lesson, we will learn about geometric sequences and series. A geometric sequence, which is also called a geometric progression, is a sequence where each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number known as the common ratio (r). We will use the lowercase r to represent our common ratio.

Finding the Common Ratio

To find r, our common ratio, we can use the following formula: $$r=\frac{a_{n + 1}}{a_{n}}$$ In other words, we can grab two terms that are next to each other in a geometric sequence and divide the rightmost term by the leftmost term to get the common ratio. Let's look at an example.
Example #1: Find the common ratio. $$-4, 24, -144, 864,...$$ Let's pick a1, which is -4, and a2, which is 24, and plug into the formula: $$r=\frac{a_{n + 1}}{a_{n}}$$ $$r=\frac{24}{-4}=-6$$

Finding the nth term

Additionally, we may be asked to find the nth term of a geometric sequence or to find the formula for the general term an. To perform this task, we can use the following formula: $$a_{n}=a_{1}\cdot r^{n - 1}$$ Let's look at an example.
Example #2: Find a12 and an. $$a_{1}=-1, r=3$$ To find a12, let's plug into our formula: $$a_{n}=a_{1}\cdot r^{n - 1}$$ $$a_{12}=-1 \cdot 3^{11}$$ $$a_{12}=-177{,}147$$ To find the formula for the general term or an, we plug in for a1 and r: $$a_{n}=-1 \cdot 3^{n - 1}$$ $$a_{n}=-3^{n - 1}$$

Geometric Series

We know that a series is the sum of the terms of a sequence. When we need to find the sum of the first n terms of a geometric sequence, we can use the following formula: $$S_{n}=\frac{a_{1}(1 - r^n)}{1 - r}$$ Let's look at an example.
Example #3: Evaluate each series. $$\sum_{i=1}^{8}2 \cdot (-4)^{i - 1}$$ Let's plug into our formula: $$S_{n}=\frac{a_{1}(1 - r^n)}{1 - r}$$ $$S_{8}=\frac{2(1 - (-4)^8)}{1 - (-4)}$$ $$S_{8}=\frac{2(1 - 65{,}536)}{1 + 4}$$ $$S_{8}=\frac{2 \cdot -65{,}535}{5}$$ $$S_{8}=2 \cdot \frac{-65{,}535}{5}$$ $$S_{8}=2 \cdot -13{,}107$$ $$S_{8}=-26{,}214$$

Skills Check:

Example #1

Find the common ratio. $$2, -10, 50, -250,...$$

A
$$r=-1$$
B
$$r=-5$$
C
$$r=-3$$
D
$$r=5$$
E
$$r=3$$

Example #2

Find a11 and an. $$a_{1}=-1, r=2$$

A
$$a_{11}=1024$$ $$a_{n}=2^{n - 1}$$
B
$$a_{11}=388$$ $$a_{n}=4^{n - 3}$$
C
$$a_{11}=256$$ $$a_{n}=(-2)^{n - 1}$$
D
$$a_{11}=-1024$$ $$a_{n}=-2^{n - 1}$$
E
$$a_{11}=568$$ $$a_{n}=(-1)^{2n}$$

Example #3

Evaluate each series. $$\sum_{i=1}^{7}(-2)^{i - 1}$$

A
$$105$$
B
$$18$$
C
$$21$$
D
$$43$$
E
$$33$$