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Last updated on 12 September 2025
The idea of a common difference is important in understanding sequences, especially arithmetic progressions. In daily life, we can see this concept in action during annual events. For a newborn, each birthday occurs exactly one year after the previous, making the common difference one year.
In an arithmetic sequence, the common difference is known as the constant value added to (or subtracted from) each term to get the next one. This difference is constant throughout the sequence. To determine the common difference, we need to subtract any term from the term that follows it. For example, in the sequence 3, 6, 9, 12, the common difference is 3. This is because 6 − 3 = 3, 9 − 6 = 3, and so on. The common difference is generally represented by the letter d.
In an arithmetic sequence, the common difference is a constant that is added to the previous term to get the next one. The formula for common difference is:
d = an − an-1
Where:
To find the common difference in an Arithmetic Progression (AP), use the formula:
d = an − an-1
Where:
Example 1: Increasing AP
Given sequence: 3, 6, 9, 12, 15, ...
Using the formula:
The common difference d is 3, signifying an increasing arithmetic progression.
Example 2: Decreasing AP
We have a sequence: 20, 16, 12, 8, 4, ...
Using the formula:
After using the formula, we find the common difference d = −4, indicating a decreasing arithmetic progression.
Example 3: Constant AP
We have Sequence: 5, 5, 5, 5, 5, ...
Using the formula:
In an arithmetic progression, d is the common difference. It is defined as the constant value which is added to every term to find the next value.
The common difference in an AP is calculated using the formula:
d = an − an-1
Where:
an is the nth term
an-1 is the (n − 1)th term
Here are examples to know Common differences can be positive, negative, or zero in a better way:
Sequence: 2, 4, 6, 8, 10, ...
d = 4 − 2 = 2
In a positive common difference, the sequence increases by 2 every time.
Sequence: 10, 7, 4, 1, −2, …
d = 7 − 10 = −3
In the negative common difference given above, the sequence decreases by 3.
Sequence: 5, 5, 5, 5, 5, ...
d = 5 − 5 = 0
In zero common difference, all the terms are equal. This means that d in a zero common difference will always be zero.
The common difference is a core concept in arithmetic progressions, where each term increases or decreases by a fixed amount. This is a powerful idea that finds practical applications in various aspects of daily life. Some of these applications are given below:
It is possible for students to make mistakes when solving problems involving common differences. However, with enough practice and attentiveness, we can avoid such mistakes. Take a look at the mistakes and solutions given below to get a better understanding of the types of mistakes we could make while dealing with common difference.
Find the common difference in the sequence 20, 16, 12, 8, 4?
In the given sequence 20, 16, 12, 8, 4, the common difference is −4.
To get the common difference, subtract the following terms: 16 − 20 = −4, 12 − 16 = −4, and so on.
The difference is constant, resulting in a common difference of −4.
Find the common difference in the sequence 5, 10, 15, 20, 25?
In the above sequence 5, 10, 15, 20, 25, the common difference is 5.
Now we will be subtracting the given terms: 10 − 5 = 5, 15 − 10 = 5, and so on.
The continuous difference presents a common difference of 5.
Find the common difference in the sequence 10, 10, 10, 10?
The common difference in the sequence 10, 10, 10, 10 is 0.
Subtract consecutive terms: 10 − 10 = 0, and so on.
As we see, the difference is zero, so the common difference will be 0.
Find the common difference in the sequence 1, 2, 3, 4, 5?
The common difference in the sequence 1, 2, 3, 4, 5 is 1.
Now we will be subtracting the provided terms: 2 – 1 = 1, 3 – 2 = 1, and so on.
The continuous difference simplifies to a common difference of 1.
Find the common difference in the sequence 7, 4, 1, -2, -5?
The common difference in the sequence 7, 4, 1, –2, –5 is –3.
Now we have to subtract consecutive terms: 4 – 7 = –3, 1 – 4 = –3, and so on.
The consistent difference gives a common difference of –3.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.