Contents

- Definition
- Sequences and the Arithmetic Sequence
- Positive Common Difference
- Negative Common Difference
- Common Difference of a Finite and Infinite Arithmetic Sequence
- Importance of Common Differences
- Finding the mth Term of an Arithmetic Sequence
- Example of Finding the Common Difference in an Arithmetic Sequence

# Common Difference|Definition & Meaning

## Definition

A **Common Difference** is a number added or subtracted to a arithmetic sequence’s term to obtain the term incoming in the **sequence**. As this difference is **common** in all the terms of an** arithmetic** sequence, it is known as the common difference. It is denoted by **d**.

## Sequences and the Arithmetic Sequence

A set of **numerals** placed in a definite order is known as a sequence. For example, **1**,**3**,**6**,**10**,**15** is a **sequence**.

There are many types of sequences but the **arithmetic sequence** deals with the concept of common difference. To understand the **common difference**, one must know about the arithmetic sequence.

An **Arithmetic Sequence** or **progression** is an ordered sequence arranged with a **common difference** between its two **consecutive** terms.

## Positive Common Difference

An arithmetic sequence with a positive common difference is shown in** figure 1**. It has the first term as **6**, the second as **12**, and so on. The **common difference d** for this sequence is **+6**.

## Negative Common Difference

The arithmetic sequence shown in **figure 2** has a negative **common difference** of **-4**. Note that the terms of the arithmetic sequence are in **descending** order.

## Common Difference of a Finite and Infinite Arithmetic Sequence

The **arithmetic sequence** can be finite or infinite. For example, the arithmetic sequence **3**,** 7**,** 11**, **15, 19** is **finite** as it has an ending term of **19**. It has a finite number of terms.

On the other hand, the arithmetic sequence **3**,**6**,**9**,**12**,… is an **infinite** sequence as the **dots** indicate that the terms will go to **infinity**.

**Figure 3** shows a finite and an infinite arithmetic sequence. Both have a **common difference** of **+2**.

## Importance of Common Differences

Common difference plays an **essential** role in an arithmetic sequence. Without a common difference, the sequence won’t even be **arithmetic!**

Also, it plays a crucial role in finding the **next **term or **any term** of the **arithmetic sequence**.

## Finding the mth Term of an Arithmetic Sequence

An **arithmetic sequence** consists of many terms. The following steps are required to find the **mth term** of an arithmetic sequence.

### Knowing the First Term

Knowing the first term is essential for computing any arithmetic term. For example, the **first term** in the arithmetic sequence **6**, **10**,** 14**, **18** is **6**. It is denoted by $b_{1}$.

### Knowing the Common Difference

The common difference** d** can be calculated by** subtracting** the **second** term from the** first** term. The value of **d** would be the same by subtracting the** third** term from the **second** term.

The mathematical **formula** for common difference can be generalized as follows:

d = $b_{m+1}$ – $b_{m}$

Where $b_{m+1}$ is the term **next** to $b_{m}$.

Without knowing the **common difference**, the mth term cannot be found.

Similarly, the next term $b_{m+1}$ can be calculated if the **previous term** b_{m} and the common difference **d** are known. The **formula** becomes:

$b_{m+1}$ = $b_{m}$ + d

### Calculating the mth Term

The following **mathematical formula** is used to find the mth term of the **arithmetic** sequence.

$b_{m}$ = $b_{1}$ + ( m – 1 ).d

Here, $b_{m}$ represents the** mth** **term**. The **m** indicates the **position** of the term in the **sequence**.

For example, the **tenth term** is required in the sequence **6**, **10**, **14**, **18** so the value of **m** will be **10**. The **first** term $b_{1}$ of the sequence is **6**.

The common difference can be calculated as follows:

d = $b_{m+1}$ – $b_{m}$

d = $b_{2}$ – $b_{1}$

d = 10 – 6

d = 4

By putting the values of **$b_{1}$**, **m**, and **d**, the tenth term of the sequence is calculated as follows:

$b_{10}$ = 6 + ( 10 – 1 )4

$b_{10}$ = 6 + ( 9 )4

$b_{10}$ = 6 + 36

$b_{10}$ = 42

The **mth term** can also be calculated by only knowing the **common difference**.

In the above sequence, **6**, **10**, **14**, **18** the common difference **4** is **added** till the** tenth** position of the sequence to find the tenth term. It is shown in **figure 4**.

## Example of Finding the Common Difference in an Arithmetic Sequence

Which of the following sequences is **arithmetic**? If arithmetic, which has a **positive** or a **negative** common difference?

a) 22, 19, 16, 13, 10, 7

b) 30, 34, 38, 42, 46, 50

c) 2, 4, 7, 9, 12

d) 4, 8, 16, 32, 64

### Solution

**(a) 22, 19, 16, 13, 10, 7**

First, the **difference** between the **first** term and the **second** term needs to be calculated. It is

$d_{1}$ = $b_{2}$ – $b_{1}$ = 19 – 22 = -3

Next, the **difference** between the **second** and the **third** term is calculated.

$d_{2}$ = $b_{3}$ – $b_{2}$ = 16 – 19 = -3

As $d_{1}$ is equal to $d_{2}$, the sequence is **arithmetic** with a **negative** common difference of **d = -3**.

**(b) 30, 34, 38, 42, 46, 50**

The **first** term is subtracted from the **second** term to find the difference $d_{1}$ as follows:

$d_{1}$ = $b_{2}$ – $b_{1}$ = 34 – 30 = +4

To check whether this **difference** is common, the **second** term is subtracted from the **third** term as follows:

$d_{2}$ = $b_{3}$ – $b_{2}$ = 38 – 34 = 4

As $d_{1}$ and $d_{2}$ are the **same**, the sequence has a **positive** common difference of **+4**. Hence, the sequence is **arithmetic**.

**(c) 2, 4, 7, 9, 12**

Calculating $d_{1}$ and $d_{2}$ to as follows:

$d_{1}$ = $b_{2}$ – $b_{1}$ = 4 – 2 = +2

$d_{2}$ = $b_{3}$ – $b_{2}$ = 7 – 4 = +3

As $d_{1}$ is not equal to $d_{2}$, there is **no common difference** in the sequence. The sequence is **not arithmetic**.

**(d) 4, 8, 16, 32, 64**

Subtracting the first from the second term gives **8 – 4 = 4**. Also, the subtracting the second from the third term gives **16 – 8 = 8**.

Both the differences are **not equal**, hence the sequence is **not arithmetic**.

The given sequence does have a **common ratio r** which is **2**. It means that the first term **4** is **multiplied** by **2** to get the second term **8**.

Similarly, the **third** term is obtained by **multiplying** the **second** term by **2**. So, the sequence is **geometric**.

*All the images are created using GeoGebra.*