Arithmetic is the root of Number theory, it involves dealing with numbers and doing various operations on numbers, including addition, subtraction, multiplication, division, and so on. Arranging those numbers in a manner that creates a pattern is known as a progression. Let’s learn more about them,

### Progressions

Progressions are numbers arranged in a particular order such that they form a predictable order. It means that from that series we can predict the next numbers of that series or sequence. For example 2, 4, 6, 8, 10,… this is a series for even Natural numbers. Progressions can be Arithmetic, geometric, harmonic, and Fibonacci. Let’s learn in detail about Arithmetic progression.

Arithmetic Progression is the sequence of the numbers where the difference between any of the two consecutive numbers is the same throughout the sequence. It can also be the common difference of that series.

### Finding the common difference

A common difference is a difference between two consecutive numbers in the arithmetic sequence. It is simply calculated by taking the difference between the second term and the first term in the arithmetic sequence or the difference between the third term and the second term or any of the two consecutive numbers in the sequence. Common Differences can be positive as well as it can be negative. The common difference is denoted as “d”. Suppose the Arithmetic progression is represented as

**a _{1}, a_{2}, a_{3},… a_{n}**

The common difference will be given as,

**d = a _{2}– a_{1} = a_{3}– a_{2} = a_{n}– a_{n-1}**

### Sample Questions

**Question 1: Find the Common difference of the Arithmetic progression- 2, 5, 8, 11, 14…**

**Solution:**

Arithmetic Sequence : 2, 5, 8, 11, 14

,…Common Difference = Second Term – First Term

= 5 – 2

= 3

Now lets take third and the second term

Common Difference = Third Term – Second Term

= 8 – 5

= 3

Here, also the common difference is same. It proves that 3 is the common difference of

2, 5, 8, 11, 14,...arithmeticsequence.

**Question 2: Find the Common difference of the Arithmetic sequence- 5, 5, 5, 5,…**

**Solution:**

It can also be the case where the common difference is 0. It will be the case when the arithmetic sequence will be of all same numbers.

Arithmetic Sequence : 5, 5, 5, 5, 5

,….Common Difference = Second Term – First Term

= 5 – 5

= 0

Here, you can see that the common difference is 0 for 5, 5, 5, 5, 5

,…arithmetic sequence

**Question 3: Find the Common difference of the Arithmetic progression- 1/2, 3/2, 5/2, 7/2,…**

**Solution:**

Arithmetic Sequence :

1/2, 3/2, 5/2, 7/2,…Common Difference = Second Term – First Term

= 3/2 – 1/2

= 1

Let’s verify by taking the fourth term and the third term

Common Difference = Fourth Term – Third Term

= 7/2 – 5/2

= 1

The common difference for the 1/2, 3/2, 5/2, 7/

2,…arithmetic sequence is 1

**Question 4: Find the Common difference of the Arithmetic progression- -3, -1, 1, 3, 5,…**

**Solution:**

Arithmetic Sequence :

–3, -1, 1, 3,5,…Common Difference = Second Term – First Term

= (-1) – (-3)

= -1 + 3

= 2

Let’s verify by taking the fourth term and the third term

Common Difference = Fourth Term – Third Term

= 3 – 1

= 2

The common difference for the

-3, -1, 1, 3, 5,… arithmetic sequence is 2