Last updated on July 4th, 2025
The idea of a common difference is important in understanding sequences, especially arithmetic progressions. In daily life, we can observe this concept unfolding in annual events. For a newborn, the next birth anniversary will be exactly one year later and the common difference will be 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 remains the same across 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:
Types of Arithmetic Sequences Based on the Common Difference:
To find the common difference in an Arithmetic Progression (AP), we have to use this 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, ...
Now we will be using the formula:
After using the formula, we get the common difference d is −4, meaning a decreasing arithmetic progression.
Example 3: Constant AP
We have Sequence: 5, 5, 5, 5, 5, ...
Now we will be 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.
Positive: When d is positive, the sequence increases.
Negative: The sequence decreases if d is negative.
Zero: If d equals 0, then all the terms will be the same.
We can use the below mentioned formula to calculate the common difference in an AP.
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:
Positive Common Difference
Sequence: 2, 4, 6, 8, 10, ...
d = 4 − 2 = 2
In a positive common difference, the sequence increases by 2 every time.
Negative Common Difference
Sequence: 10, 7, 4, 1, -2, …
d = 7 − 10 = -3
In the negative common difference given above, the sequence decreases by 3.
Zero Common Difference
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:
Birthday Candles - Every year on your birthday, you add one more candle to your cake. If you had 1 candle at age 1, 2 at age 2, and so on, the number of candles increases by 1 each year. This forms an arithmetic progression where the common difference is 1.
Saving Money - If you save ₹100 on the first day, ₹200 on the second day, ₹300 on the third day, and so on, you're increasing your savings by ₹100 each day. This is an arithmetic progression with a common difference of ₹100.
Reading Pages - If you read 5 pages of a book on the first day, 10 pages on the second day, 15 pages on the third day, and so on, you're increasing the number of pages you read by 5 each day. This follows an arithmetic progression with a common difference of 5.
Juice Bottles - If you start with 2 bottles of juice and drink 1 bottle each day, the number of bottles remaining decreases by 1 each day. This repeating pattern creates an arithmetic progression.
Arranging Blocks - When you stack blocks, each layer can have more blocks than the one below it. If the first layer has 1 block, the second has 3, the third has 5, and so on, the number of blocks increases by 2 in each layer. This pattern creates an arithmetic progression where the common difference would be 2.
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.
Now we need to subtract the mentioned 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.