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.

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 a### Finding the n

In some cases, we will be asked to find the n

Example #2: Find a

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. $$S_{n}=\frac{n}{2}[2a_{1}+ (n - 1)d]$$ $$S_{40}=\frac{40}{2}[2(2) + 7(40 - 1)]$$ $$S_{40}=20[4 + 7(39)]$$ $$S_{40}=20(277)$$ $$S_{40}=5540$$

### 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 a

_{n + 1}, while the one on the left will serve as the a_{1}. Let's choose a_{1}and a_{2}, 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 n^{th} Term of an Arithmetic Sequence

In some cases, we will be asked to find the n^{th}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 a

_{22}and a_{n}. $$28, 31, 34, 37$$ $$d=31 - 28=3$$ Now, let's plug into our formula: $$a_{n}=a_{1}+ d(n - 1)$$ a_{n}is what we want to find, here this is a_{22}: $$a_{22}=28 + 3(22 - 1)$$ $$a_{22}=28 + 3 \cdot 21$$ $$a_{22}=28 + 63$$ $$a_{22}=91$$ How do we find the formula for the general term a_{n}? We just plug in for a_{1}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 $$2, 0, -2, -4$$

Please choose the best answer.

A

$$d=-3$$

B

$$d=-2$$

C

$$d=2$$

D

$$d=\frac{1}{2}$$

E

$$d=4$$

Example #2

Find a_{20} and a_{n}. $$23, 27, 31, 35,...$$

Please choose the best answer.

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

Please choose the best answer.

A

$$177$$

B

$$200$$

C

$$151$$

D

$$354$$

E

$$552$$

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