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 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:

• aₙ is the nth term of the sequence.
   
• aₙ₋₁ represents the term immediately preceding aₙ

Types of Arithmetic Sequences Based on the 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), we have to use this formula:

d = aₙ − aₙ₋₁

Where:

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

Example 1: Increasing AP

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

• a₁ = 3
   
• a₂ = 6
   
• a₃ = 9

Using the formula:

• d = a₂ − a₁ = 6 − 3 = 3
   
• d = a₃ − a₂ = 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

Now we will be using the formula:

• d = a₂ − a₁ = 16 − 20 = −4
   
• d = a₃ − a₂ = 12 − 16 = −4

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, ...

• a₁ = 5
   
• a₂ = 5
   
• a₃ = 5

Now we will be using the formula:

• d = a₂ − a₁ = 5 − 5 = 0
   
• d = a₃ − a₂ = 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: 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.