What is the common difference of an Arithmetic Progression?

Common Difference of an Arithmetic Progression is the constant value difference between the next term and the previous term of a progression. It is denoted by d in mathematics. The common difference of an Arithmetic Progression can be positive, negative, or zero.

Formula for calculating the common difference in Arithmetic progression is d = a_n – a (n – 1), where a_n is n^th term of A.P.

We can find common difference of an A.P. is constant for every pair of number. If there is two common difference present in sequence then it is not an Arithmetic progression. The terms of an arithmetic progression can be algebraically represented as a, a + d, a + 2d, a + 3d, a + 4d, a + 5d . . . where a is the first term and d is the common difference.

What is 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), First term (a), and n^th Term (a_n). Other then these, we can also calculate Sum of the first n terms (S_n) for any A.P.

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, a₃, a₄, and so on. So,

a₂ – a₁ = d

a₃ – a₂ = d

a₄ – a₃ = d and so on.

Example: 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)d

where,  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)d

Where, 

• a = First term of AP
• d = Common difference
• n = number of terms
• a_n = n^th 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)d

a₃₀ = 3 + (30− 1)2

a₃₀ = 3 + 58

a₃₀ = 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)d

First-term, a =1

Common difference, d = 2-1 = 1

Therefore, a_n = 1+(10-1)1 = 1+9 = 10