Lesson Objectives
• Learn how to find the common difference of an arithmetic sequence
• Learn how to find a specific term and formula for an arithmetic sequence
• Learn how to evaluate an arithmetic series

## What is an Arithmetic Sequence?

In this lesson, will learn about arithmetic sequences and series. A sequence in which each term after the first is obtained by adding some fixed number to the previous term is known as an arithmetic sequence or an arithmetic progression. The fixed number is known as the common difference and is usually expressed with a lowercase d.

### Finding the Common Difference

To find the common difference for an arithmetic sequence, we can use a simple formula: $$d=a_{n + 1}- a_{n}$$ Let's look at an example.
Example #1: Find the common difference. $$29, 39, 49, 59,...$$ We can choose any two numbers that are next to each other. The one on the right has the higher index value, it will serve as the an + 1, while the one on the left will serve as the a1. Let's choose a1 and a2, which gives us 29 and 39. We plug into our formula: $$d=a_{n + 1}- a_{n}$$ $$d=a_2 - a_1$$ $$d=39 - 29=10$$ Our common difference is 10.

### Finding the nth Term of an Arithmetic Sequence

In some cases, we will be asked to find the nth term of an arithmetic sequence and give the general formula. To accomplish this task, we use the following formula: $$a_{n}=a_{1}+ (n - 1)d$$ Let's look at an example.
Example #2: Find a22 and an. $$28, 31, 34, 37$$ $$d=31 - 28=3$$ Now, let's plug into our formula: $$a_{n}=a_{1}+ d(n - 1)$$ an is what we want to find, here this is a22: $$a_{22}=28 + 3(22 - 1)$$ $$a_{22}=28 + 3 \cdot 21$$ $$a_{22}=28 + 63$$ $$a_{22}=91$$ How do we find the formula for the general term an? We just plug in for a1 and d: $$a_{n}=28 + 3(n - 1)$$ $$a_{n}=28 + 3n - 3$$ $$a_{n}=25 + 3n$$

### Sum of the First n Terms of an Arithmetic Sequence

Recall that a series is sum of the terms of a sequence. When we sum the terms of an arithmetic sequence, this is known as an arithmetic series. We have a very useful formula that allows us to find the sum of the first n terms of an arithmetic sequence. $$S_{n}=\frac{n}{2}(a_{1}+ a_{n})$$ $$S_{n}=\frac{n}{2}[2a_{1}+ (n - 1)d]$$ The first formula will be used when the first and last terms are known, otherwise the second formula is used. Let's look at an example.
Example #3: Evaluate each arithmetic series. $$a_{1}=2, d=7, n=40$$ Let's use the second formula since we don't know the last term. $$S_{n}=\frac{n}{2}[2a_{1}+ (n - 1)d]$$ $$S_{40}=\frac{40}{2}[2(2) + 7(40 - 1)]$$ $$S_{40}=20[4 + 7(39)]$$ $$S_{40}=20(277)$$ $$S_{40}=5540$$

#### Skills Check:

Example #1

Find the common difference $$2, 0, -2, -4$$

A
$$d=-3$$
B
$$d=-2$$
C
$$d=2$$
D
$$d=\frac{1}{2}$$
E
$$d=4$$

Example #2

Find a20 and an. $$23, 27, 31, 35,...$$

A
$$a_{20}=94, a_{n}=19 + 2n$$
B
$$a_{20}=100, a_{n}=2 - 5n$$
C
$$a_{20}=9, a_{n}=11 + 5n$$
D
$$a_{20}=99, a_{n}=19 + 4n$$
E
$$a_{20}=16, a_{n}=20 + 3n$$

Example #3

Evaluate each series $$a_{1}=13, d=3, n=12$$

A
$$177$$
B
$$200$$
C
$$151$$
D
$$354$$
E
$$552$$

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