# What is the common difference of an AP?

An ordered list of numbers is called a sequence. Each number of the sequence is called a term. A sequence is denoted as a_{1}, a_{2}, a_{3}, a_{4}, …. a_{n }where, a_{1} is the first term, a_{2 }is the second term… a_{n }is the nth term. A finite sequence consists of a finite list of numbers such as for example {2, 4, 8, 16, 32} is a finite sequence whereas an infinite sequence consists of an infinite list of numbers such as for example {3,7,11,15,…}.

A sequence with a specific Pattern is called **Progression**. The sum of the terms of a sequence is called a **Series**. The corresponding Series of the above-defined sequence is a_{1 }+ a_{2} + a_{3} + a_{4 }+ …. + a_{n}. The series can be finite or infinite depending if the sequence is finite or infinite.

### Arithmetic Sequence

A sequence is called an arithmetic sequence if the difference between the two terms is constant. Example: let us look at the sequence {-6, -3, 0, 3, 6,….}. While observing the above example it can be easily identified that adding 3 for the first term of the sequence to get the second term and similarly adding 3 for the second term to get the third term and so on. So let us assume a = -6 and d = 3 (constant) we can define the above sequence as {a, a+d, a+2d, a+3d,….}. Therefore, the rule for defining an arithmetic sequence will be as,

**a _{n }= a + d(n – 1) **

Where a is the first term and d is the common difference

### What is the common difference of an AP?

It is now known that each term of an arithmetic progression is obtained by adding a constant to the previous term except the first term of the sequence. This constant is called “Common Difference”. The common difference of an arithmetic sequence is defined as the constant difference between the two consecutive terms of a sequence. The formula for obtaining the common difference is,

**d = a _{n }– a_{n-1}**

**Properties of Common difference**

- If a constant is added to each term of an A.P, the resulting sequence is also an A.P
- If a constant is subtracted from each term of an A.P, the resulting sequence is also an A.P
- If a constant is multiplied by each term of an A.P, the resulting sequence is also an A.P
- If a non zero constant is divided by each term of an A.P, the resulting sequence is also an A.P

**Example of Common Difference**

**Example 1 **Let’s look at the following arithmetic sequence

100, 95, 90, 85, 80, 75, ….

Let’s find the common difference as below,

a_{2 }– a_{1 }= 95 – 100 = -5

a_{3} – a_{2} = 90 – 95 = -5

a_{4 }– a_{3 }= 85 -90 = -5

So, here the common difference is **-5** which is negative. This type of arithmetic progression is called Decreasing A.P or Reducing A.P.

**Example 2 **Let’s look at the following example,

3, 8, 13, 18, 23, ….

let’s find the common difference using the formula d = a_{n} – a{n – 1}

n = 2; d = a_{2} – a_{1} = 8 – 3 = 5

n = 3; d = a_{3} – a_{2 }= 13 – 8 = 5

n = 4; d = a_{4} – a_{3} = 18 – 13 = 5

Therefore, the common difference in this example is **5** which means the ap is increasing. This type of A.P is called Increasing A.P.

### Sample Problems

**Question 1: Find the common difference, **

**3, 3, 3, 3, 3, …**

**Solution:**

let’s find the common difference,

n = 2; d = a2 – a1 = 3 – 3 = 0

n = 3; d = a3 – a2 = 3 – 3 = 0

n = 4; d = a4 – a3 = 3 – 3 = 0

Therefore, the common difference here is

0. Even 0 is considered as constant. So, this type of sequence is also considered as Arithmetic progression.

**Question 2: Find the common difference,**

**2, 4, 6, 8, 10,12 ,….**

**Solution:**

Let’s consider the sequence, 2, 4, 6, 8, 10,12 ,….

Let’s multiply each term of the above sequence by 2. The resulting sequence will be,

4, 8, 12, 16, 20, …..

Now,

n = 2; d = a

_{2 }– a_{1}= 8 – 4 = 4n = 3; d = a

_{3}– a_{2 }= 12 – 8 = 4n = 4; d = a

_{4}– a_{3}= 16-12 = 4Therefore, the common difference is a constant 4 and it satisfies the property number 3 as discussed above.

**Question 3: Find the common difference, 3, 6, 9, 12, 15,…**

**Solution:**

Let’s consider the sequence 3, 6, 9, 12, 15,…

Let’s divide the above sequence by 3 . The resulting sequence will be

1, 2, 3, 4, 5,….

Now,

n = 2; d = a

_{2 }– a_{1}= 2 -1 = 1n = 3; d = a

_{3}– a_{2}= 3 – 2 = 1n = 4; d = a

_{4 }– a_{3}= 4 – 3 = 1Therefore, the common difference is a constant

1and it satisfies the property number 4 as discussed above.

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