Common Difference

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:

\(d = a_n - a_{n-1} \)

Where:

• a_n is the nth term of the sequence.
   
• a_n-1 represents the term immediately preceding a_n.

Types of Arithmetic Sequences Based on Common Difference:

• Increasing Sequence: If d is positive, every term is greater than the previous one.
   
• Decreasing Sequence: If d is negative, each term will be less than the previous term.
   
• Constant Sequence: If d is zero, all terms will be equal.

To find the common difference in an arithmetic progression (AP), use the formula:

\(d = a_n - a_{n-1} \)

Where:
 

• a_n is the nth term of the sequence.
   
• a_n-1 is the (n − 1)th term, or the term preceding a_n.

Example 1: Increasing AP

Given sequence: 3, 6, 9, 12, 15, ...
 

• a₁ = 3
   
• a₂ = 6
   
• a₃ = 9

Using the formula:
 

• \(d = a_2 − a_1\) = 6 − 3 = 3
   
• \(d = a_3 − a_2\) = 9 − 6 = 3

The common difference d is 3, signifying an increasing arithmetic progression.

Example 2: Decreasing AP

We have a sequence: 20, 16, 12, 8, 4, ...

• a₁ = 20
   
• a₂ = 16
   
• a₃ = 12

Using the formula:

• \(d = a_2 − a_1\) = 16 − 20 = −4
   
• \(d = a_3 − a_2\) = 12 − 16 = −4

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, ...

• a₁ = 5
   
• a₂ = 5
   
• a₃ = 5

Using the formula:

• \(d = a_2 − a_1\) = 5 − 5 = 0
   
• \(d = a_3 − a_2 \)= 5 − 5 = 0

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: If d > 0, the sequence increases.
   
• Negative: If d < 0, the sequence decreases.
   
• Zero: If d = 0, all terms are equal.

The common difference in an AP is calculated using the formula:

\(d = a_n − a_n-1\)

Where:

a_n is the n^th term

a_n-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.

Understanding the common difference is key to solving arithmetic sequences efficiently. Regular practice and recognizing patterns can make mastering it easier.

• Look at the sequence carefully and find the constant difference between consecutive terms.
   
• Subtract any term from the next term in the sequence to calculate the common difference quickly.
   
• Ensure the difference is the same throughout the sequence to confirm it is truly arithmetic.
   
• Use the arithmetic sequence formula \(a_n​=a_1​+(n−1)d\) to find missing terms or the common difference.
   
• Practice with examples like salaries, seat arrangements, or plant growth to reinforce understanding of common difference.

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:

• Salary increment: When employees receive a fixed annual raise, the difference in salary each year is a common difference. For example, If a salary increases by ₹2000 every year, the common difference is ₹2000.
   
• Seating arrangements: In theaters or auditoriums, seats are often arranged in rows with a fixed number of additional seats in each row, forming an arithmetic pattern. For example, 10 seats in the first row, 12 in the second, 14 in the third – common difference is 2 seats.
   
• Plant growth: When plants grow by a fixed amount each week, the growth forms an arithmetic sequence. For example, A plant grows 3 cm every week; the common difference is 3 cm.
   
• Savings and investment: Regular deposits in a bank with a fixed increase each month can be modeled using a common difference. For example, depositing ₹500 more than the previous month; common difference = ₹500.
   
• Staircase design: The height of each step in a staircase is uniform, creating a common difference in the vertical distance between steps. For example, Each step rises 15 cm; common difference = 15 cm.