The common difference is the amount between each number in an arithmetic sequence. 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 that 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 that the common difference is 3.
Therefore, you can say that the formula to find the common difference of an arithmetic sequence is: d = a(n) - a(n - 1), where a(n) is the last term in the sequence, and a(n - 1) is the previous term in the sequence. 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
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Additional Activities
Common Difference: Additional Practice Problems
1. Find the common difference in the following sequence: 9, -1, -11, -21,...
2. Find the common difference in the following sequence: 0, 2, 5, 8.
3. Find the common difference in the following sequence: 4, 8, 12, 16, 20,...
4. Find the common difference in the sequence: -8, -14, -20, -26,...
Solutions
1. Looking at the terms of the sequence, we subtract each term from the term following it to get:
-21 - (-11) = -21 + 11 = -10
-11 - (-1) = -11 + 1 = -10
-1 - 9 = -10
In each case we get the same number -10. Hence, the common difference is -10.
2. Looking at the terms of the sequence, we subtract each term from the term following it to get:
8 - 5 = 3
5 - 2 = 3
2 - 0 = 2
In this case we DO NOT have the same difference between successive terms. Hence, there is NO common difference in this sequence.
3. Looking at the terms of the sequence, we subtract each term from the term following it to get:
20 - 16 = 4
16 - 12 = 4
12 - 8 = 4
8 - 4 = 4
In each case we get the same number 4. Hence, the common difference is 4.
4. Looking at the terms of the sequence, we subtract each term from the term following it to get:
-26 - (-20) = -6
-20 - (-14) = -6
-14 - (-8) = -6
In each case we get the same number -6. Hence, the common difference is -6.
Note that the common difference doesn't necessarily have to be a positive number. Some of the questions have a positive and other may have a negative common difference.
Answers & Comments
Answer:
Determining the Common Difference
The common difference is the amount between each number in an arithmetic sequence. 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 that 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 that the common difference is 3.
Therefore, you can say that the formula to find the common difference of an arithmetic sequence is: d = a(n) - a(n - 1), where a(n) is the last term in the sequence, and a(n - 1) is the previous term in the sequence. 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
To unlock this lesson you must be a Study.com Member.
Create your account
Additional Activities
Common Difference: Additional Practice Problems
1. Find the common difference in the following sequence: 9, -1, -11, -21,...
2. Find the common difference in the following sequence: 0, 2, 5, 8.
3. Find the common difference in the following sequence: 4, 8, 12, 16, 20,...
4. Find the common difference in the sequence: -8, -14, -20, -26,...
Solutions
1. Looking at the terms of the sequence, we subtract each term from the term following it to get:
-21 - (-11) = -21 + 11 = -10
-11 - (-1) = -11 + 1 = -10
-1 - 9 = -10
In each case we get the same number -10. Hence, the common difference is -10.
2. Looking at the terms of the sequence, we subtract each term from the term following it to get:
8 - 5 = 3
5 - 2 = 3
2 - 0 = 2
In this case we DO NOT have the same difference between successive terms. Hence, there is NO common difference in this sequence.
3. Looking at the terms of the sequence, we subtract each term from the term following it to get:
20 - 16 = 4
16 - 12 = 4
12 - 8 = 4
8 - 4 = 4
In each case we get the same number 4. Hence, the common difference is 4.
4. Looking at the terms of the sequence, we subtract each term from the term following it to get:
-26 - (-20) = -6
-20 - (-14) = -6
-14 - (-8) = -6
In each case we get the same number -6. Hence, the common difference is -6.
Note that the common difference doesn't necessarily have to be a positive number. Some of the questions have a positive and other may have a negative common difference.