Common Difference
Common difference is a concept used in sequences and arithmetic progressions. The celebration of people's birthdays can be considered as one of the examples of sequence in real life. In this example, the common difference between consecutive celebrations of the same person is one year.
In this article, let's learn about common difference, and how to find it using solved examples.
1.  What Is Common Difference? 
2.  Common Difference Formula 
3.  Finding Common Difference in Arithmetic Progression 
4.  Examples on Common Difference 
5.  FAQs on Common Difference 
What Is Common Difference?
An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same amount. The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference", because if we subtract (that is if you find the difference of) successive terms, you'll always get this common value. Most often, "d" is used to denote the common difference.
Consider the arithmetic sequence: 2,4,6,8,..
Here, the common difference between each term is 2.
4  2 = 2
6  4 = 2
8  6 = 2
Common Difference Formula
The common difference is the value between each successive number in an arithmetic sequence. Therefore, 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. There are two kinds of arithmetic sequence:
 Increasing arithmetic sequence
 Decreasing arithmetic sequence
Some sequences are made up of simply random values, while others have a fixed pattern that is used to arrive at the sequence's terms. The arithmetic sequence (or progression), for example, is based upon the addition of a constant value to reach the next term in the sequence. The number added to each term is constant (always the same).
a_{1}, (a_{1} + d), (a_{1} + 2d), ...
The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms generates the constant value that was added. To find the common difference, subtract the first term from the second term.
Arithmetic sequences have a linear nature when plotted on graphs (as a scatter plot). The domain consists of the counting numbers 1, 2, 3, 4,5 ... (representing the location of each term) and the range consists of the actual terms of the sequence. In the graph shown above, while the xaxis increased by a constant value of one, the y value increased by a constant value of 3.
Finding Common Difference in Arithmetic Progression
Let's make an arithmetic sequence with a starting number of 2 and a common difference of 5.
 Our first term will be our starting number: 2. Our second term will be equal to the first term (2) plus the common difference (5), so our second term is 7.
 So the first two terms of our sequence are 2, 7. Our third term will be equal to the second term (7) plus the common difference (5), so our third term is 12.
 So the first three terms of our sequence are 2, 7, 12. Our fourth term will be equal to the third term (12) plus the common difference (5), so our fourth term is 17.
 So the first four terms of our sequence are 2, 7, 12, 17.
Now we are familiar with making an arithmetic sequence from a starting number and a common difference. Now, let's learn how to find the common difference of a given sequence. 2,7,12,..
 The first term here is 2; so that is the starting number.
 The second term is 7. To find the difference between this and the first term, we take 7  2 = 5. So the difference between the first and second terms is 5.
 The second term is 7 and the third term is 12. To find the difference, we take 12  7 that gives us 5 again. So the common difference between each term is 5.
The belowgiven table gives some more examples of arithmetic sequences and shows how to find the common difference of the sequence.
Arithmetic Sequence  Common Difference 'd' 
1, 6, 11, 16, 21, 26, ...  d = 5. 5 is added to each term to arrive at the next term. so ... the difference a2  a1 = 5. 
10, 8, 6, 4, 2, 0, 2, 4, 6,..  d = 2. 2 is added to each term to arrive at the next term.so ... the difference a2  a1 = 2. 
1, 1/2, 0, 1/2,..  d = ½. A ½ is added to each term to arrive at the next term.so ... the difference a2  a1 = ½. 
Related Articles on Common Difference
Check out the following pages related to Common Difference
Important Notes on Common Difference
Here is a list of a few important points related to common difference.
 An arithmetic sequence goes from one term to the next by always adding or subtracting the same amount.
 The number added or subtracted at each stage of an arithmetic sequence is called the "common difference"
 The common difference is denoted by 'd'.
Examples on Common Difference

Example 1: Determine the common difference in the given sequence: 3,0,3,6,9,12,..
Solution:
Given sequence: 3,0,3,6,9,12,..
The common difference is the distance between each number in the sequence. Notice that each number is 3 away from the previous number.
12 − 9 = 3
9 − 6 = 3
6 − 3 = 3
3 − 0 = 3
0 −(−3) = 3
Thus, the common difference is 3.

Example 2: What is the common difference in the following sequence?
3,11,19,27,35..
Solution:
Given sequence: 3,11,19,27,35..
To find the common difference, simply subtract the first term from the second term, or the second from the third, or so on...
11−3 = 8
19−11 = 8
Clearly, each time we are adding 8 to get to the next term. Thus, the common difference is 8.
FAQs on Common Difference
What Is the Common Difference of Any Arithmetic Sequence?
An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference between two consecutive terms is called the common difference.
How To Find the Common Difference?
The common difference is the value between each number in an arithmetic sequence.
 Step 1: Take any two consecutive terms.
 Step 2: Find their difference, 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.
Can the Common Difference of AP Be Negative?
Yes, the common difference of an arithmetic progression can be positive, negative, or even zero. In a decreasing arithmetic sequence, the common difference is always negative as such a sequence starts out negative and keeps descending.
Can the Common Difference in AP Be Zero?
Yes. The common difference in an arithmetic progression can be zero. As per the definition of an arithmetic progression (AP), a sequence of terms is considered to be an arithmetic sequence if the difference between the consecutive terms is constant. Thus, an AP may have a common difference of 0.
And since 0 is a constant, it should be included as a common difference, but it kinda feels wrong for all the numbers to be equal while being in an arithmetic progression.
What Is the Symbol of Common Difference?
The common difference is the value between each term in an arithmetic sequence and it is denoted by the symbol 'd'.
What If the Common Difference Is Not Constant?
In a sequence, if the common difference of the consecutive terms is not constant, then the sequence cannot be considered as arithmetic.
What Is the Common Difference Formula?
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.
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