What is the common difference of four terms in an AP?

Arithmetic has most likely the longest history at the time. It is a method of computation that has been used since ancient times to derive definite values for routine calculations such as measurements, labelling, and other day-to-day calculations. The phrase derives from the Greek word “arithmos,” which translates as “numbers.”

Arithmetic is the basic branch of mathematics that studies numbers and the characteristics of classical operations such as addition, subtraction, multiplication, and division.

Besides the basic operations of addition, subtraction, multiplication, and division, arithmetic also includes advanced computations such as percentage, logarithm, exponentiation, and square roots, among others. Arithmetic is the study of numerals and their traditional operations.

Arithmetic Progression

Arithmetic Progression (AP) is a numerical sequence in which the difference between any two subsequent numbers is always the same.

For Example: The numbers 1, 2, 3, 4, 5, 6,… are in Arithmetic Progression, which has a common difference (d) between two subsequent terms (say, 1 and 2) equal to 1. (2 – 1).

Even for odd and even numbers, there is a common difference between two successive terms. Common difference (d), the nth term (an), and the sum of the first n terms (S_n) are the three main terms in AP. All three terms indicate AP’s properties.

Let’s look at the common difference in more detail:

In other words, arithmetic progression is “a mathematical sequence in which the difference between two subsequent terms is always a constant.”

General form of Arithmetic progression is  

a, a + d, a + 2d, a+3d……… a + (n – 1) d

What is the common difference of four terms in an A.P.?

In arithmetic progression, the common difference is indicated by d. It’s the distinction between the next term and the one preceding it. It is always constant or the same in arithmetic progression. In other words, if the common difference in a given sequence is constant, we may conclude that this is A.P. or the sequence is as follows: a₁, a₂, a₃, a₄, and so on. In other words, d represents the common difference in the arithmetic progression. The difference between the succeeding and preceding terms. For arithmetic progression, the common difference is always constant or the same.

To find common difference (d) = (a_n + 1 – a_n) or d = (a_n – a_n – 1)

If the common difference is positive, then AP will increase . For Example 4, 8, 12, 16 , 20 , 24 ….. in this series, The difference is positive therefore , AP increases .

If the common difference is negative then AP will decrease. For Example -4, -6, -8 , -10 , -12……., here difference is negative therefore, AP decreases.

If the common difference is zero then AP will be constant. For Example 1, 1, 1, 1, 1,1,1……, here AP will be  constant.

The sequence of Arithmetic Progression will be like a₁, a₂, a₃, a₄, a₅ ……

Common difference (d) = a₂ – a₁ = d  

or, a₃ – a₂ = d

or, a₄ – a₃ = d

or, a₅ – a₄ = d and so on.

Suppose if the four terms are, a₁ = a, a₂ = a + d, a₃ = a + 2d and a₄ = a + 3d ….

Now as per the formula to find common difference =

common difference = a₄ – a₃

= a + 3d – (a+2d)

= a + 3d – a – 2d

= d

Also, common difference = a₃ – a₂

= a + 2d – (a + d)

= a + 2d – a – d

= d

Solved Examples on Common Difference of AP

Example 1: What is the common difference in the following sequence of A.P.?

3, 10,17, 24, 31…

Solution:

Given sequence: 3, 10,17, 24, 31…..

To find the common difference :

We have to subtract the first term from the second term, or the second from the third, 

Therefore, 10−3 = 7

or, 17−10 = 7

or, 24 – 17= 7

Each time we are adding 7 to get to the next term. Hence, the common difference is 7.

Example 2: Find the AP if the first term is 20 and the common difference is 5.

Solution:

As we know that an AP is ,

a, a + d, a + 2d, a + 3d, a + 4d, …

Here, a = 20 and d = 5.

a, a + d, a + 2d, a + 3d, a + 4d, …

= 20, (20 + 5), (20 + 2 x 5), (20 + 3 x 5), (20 + 4 x 5),…

= 20, 25, (20 + 10), (20 + 15), (20 + 20), …

= 20, 25, 30 , 35 , 40, …and so on.

Example 3: Find the first four terms of an AP whose first term is –3 and the common difference is –2.

Solution:

As we know that A.P. is,

a, a + d, a + 2d, a + 3d, a + 4d, …

Here, a = -3 and d = -2

a, a + d, a + 2d, a + 3d, a + 4d, …

= -3 , {-3 + (-2)}, {-3 + (2 x (-2)}, {-3 + 3 x (-2)}, {-3 + 4 x (-2)},…

= -3 , -5 , (-3 -4 ), (-3 -6 ), (-3 – 8), …

= -3,-5,-7,-9,-11 …and so on.

Example 4: Find the 20th term for the given AP: 2, 6, 10, 14, …?

Solution:

Given 2, 6, 10, 14, ……

a = 2, d = 6 – 2 = 4, n = 20

a_n = a + (n − 1)d

a₂₀  = 2 + (20− 1)4

a₂₀  = 2 + 19(4)

a₂₀  = 2 + 76 = 78

Example 5: What is the common difference in the following sequence of A.P.?

5, 10,15, 20, 25…

Solution:

Given sequence : 5, 10, 15, 20, 25…

To find the common difference :

We have to subtract the first term from the second term, or the second from the third,

Therefore , 10 − 5 = 5

or, 15−10 = 5

or, 20 – 15 = 5

Each time we are adding 5 to get to the next term. Hence, the common difference is 5.

FAQs on Common Difference

Question 1. What is the common difference of an A.P.?

Solution:

The common difference of an A.P. is defined as the difference between the next and previous terms and its preceding term. Its value is always a fixed constant and same for entire progression.

Question 2. Write the formula for calculating the sum of an arithmetic progression in terms of common difference.

Solution:

The sum of an A.P. is given by the formula:

S = n/2 [2a + (n – 1)d]

where,

• S is the sum of the series
• n is the number of terms
• a is the first term
• d is the common difference

Question 3. How does the common difference change when all the terms of an A.P. are increased by a value of 5?

Solution:

The common difference increases on increases in the value of terms and decreases on decreases in the value of terms. So, it will also increase by a value of 5.