# What is the common difference of an Arithmetic Progression?

Arithmetic Progression, (AP) is a sequence of numbers in which the difference between any two consecutive numbers is a constant value. For example, the series of numbers: 1, 2, 3, 4, 5, 6,… are in Arithmetic Progression, which has a common difference (d) between two successive terms (say 1 and 2) equal to 1 (2 -1). A common difference can be seen between two successive terms, even for odd numbers and even numbers, that is 2 is equal to. In AP, three main terms are Common difference (d), nth Term (a_{n}), Sum of the first n terms (S_{n}), All three terms represent the properties of AP. Let’s take a look at what common difference is in detail,

### Common Difference of an A.P.

The common difference is denoted by **d** in arithmetic progression. It’s the difference between the next term and the one before it. For arithmetic progression, it is always constant or the same. In a word, if the common difference is constant in a certain sequence, we can say that this is A.P. If the sequence is a_{1, }a_{2, }a_{3}, a_{4}, and so on. So,

a

_{2 }– a_{1 }= da

_{3 }– a_{2 }= da

_{4 }– a_{3 }= d and so on.

**Question: Find the common difference in the A.P. 7,11,15,19,23,…**

**Solution:**

Here, it is known that the common difference can be found by finding the difference between two consecutive terms,

11 – 7 = 4

15 -11 = 4

19 – 15 = 4

23 – 19 = 4

Other important terms that are used to explain the properties of an Arithmetic progression are,

**First Term of AP:**

The Arithmetic Progression can be written in terms of common difference (d) as:

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1)dwhere, a= first term of AP

**n-th term of AP:**

The nth term can be found by using the formula mentioned below:

a_{n}= a + (n − 1)dWhere,

a = First term of AP

d = Common difference

n = number of terms

a

_{n}= nth term

**Note:** The sequence’s behavior is based on the value of a shared difference.

- If “d” is positive, the terms will increase to positive infinity.
- If “d” is negative, the terms of the members increase to negative infinity

**Sum of n terms**

The formula for the AP sum is explained below, consider an AP consisting of “n” terms.

S = n/2 [2a + (n − 1) d]

**Sum of AP when the First and Last Term is Given**

S = n/2 (first term of AP + last term of AP)

### Solved Examples

**Question 1: Find the AP if the first term is 10 and the common difference is 4.**

**Solution: **

It is known,

a, a + d, a + 2d, a + 3d, a + 4d, …

Here, a = 10 and d = 4

= 10, (10 + 4), (10 + 2 x 4), (10 + 3 x 4), (10 + 4 x 4),…

= 10, 14, (10 + 8), (10 + 12), (10 + 16), …

= 10, 14, 18, 22, 26, …and so on.

**Question 2:** **Find the 30th term for the given AP: 3, 5, 7, 9, …**

**Solution: **

Given, 3, 5, 7, 9, ……

a = 3, d = 5 – 3 = 2, n = 30

a

_{n}= a + (n − 1)da

_{30}= 3 + (30− 1)2a

_{30}= 3 + 58a

_{30}= 61

**Question 3: Find the sum of 10 terms of AP. AP = 1,2,3,4,5,6,7,8,9,10**

**Solution: **

Given, a = 1, d = 2-1 = 1 and n = 15

S = n/2[2a + (n − 1)d]

S = 10/2[2*1+(10-1).1]

S = 5[2+9] = 5 x 11

S = 55

**Question 4: Find the nth term of AP: 1, 2, 3, 4, 5…, an, if the number of terms is 10.**

**Solution: **

n=10

a

_{n}= a+(n-1)dFirst-term, a =1

Common difference, d = 2-1 = 1

Therefore, a

_{n}= 1+(10-1)1 = 1+9 = 10

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