# Arithmetic Formulas: Definition & Examples

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• 0:00 Arithmetic Formula Defined
• 0:45 Sum Of An Arithmetic Formula
• 1:35 Determining The Common…
• 2:25 Examples
• 4:20 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

An arithmetic formula is a group of numbers with a common difference. Arithmetic formulas are sometimes called arithmetic series or arithmetic sequences. This lesson will define an arithmetic formula and give some examples.

## Arithmetic Formula Defined

Before we begin, we must first define a couple of basic terms. A sequence, or series, is a group of numbers that can be written in a particular order, or it can just be a random set. An arithmetic sequence, also known as an arithmetic formula or arithmetic series, is a group of numbers that is ordered with a specific pattern. The pattern is determined by adding a certain number to each number in the sequence to determine the next number in the sequence. The number added must be the same for each term in the sequence, and that number is called the common difference.

Arithmetic formulas have practical uses in biology, engineering and other scientific disciplines. They can be used to determine patterns in architecture and also have uses in working with finances.

## Sum of an Arithmetic Formula

You can determine the sum of an arithmetic formula by using the following equation:

S = (n/ 2) x (a1 + an)

Where:

S = the sum

n = the number of terms in the series

a1 = the first term in the series

an = the last term in the series

For example, find the sum of the following arithmetic series:

{3, 5, 7, 9, 11, 13, 15, 17}

We could add this the long way, but with longer sequences, it becomes impractical when we can just use the equation.

S = (n/ 2) x (a1 + an)

S = (8 / 2) x (3 + 17)

S = 4 x 20

S = 80

## Determining the Common Difference

The common difference is the amount between each number in an arithmetic formula. It is called common difference because it is the same, or common to, each number, and it also is the difference between each number in the sequence.

To determine the common difference, you can just subtract each number from the number following it in the sequence. For example, what is the common difference in the following sequence of numbers?

{1, 4, 7, 10}

Starting with the number at the end of the sequence, subtract the number immediately preceding it

10 - 7 = 3

Continue to subtract to ensure the pattern is the same for each number in the series.

7 - 4 = 3

4 - 1 = 3

Since the difference is the same for each set, you can say the common difference is 3.

If you subtract and find that the difference between each number in the sequence is not the same, then there is no common difference and the sequence is not arithmetic.

## Examples

1.) What is the common difference in the following sequence?

{3, 8, 13, 18}

18 - 13 = 5

13 - 8 = 5

8 - 3 = 5

The difference between each of the numbers in the sequence is 5, therefore the common difference is 5.

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