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.

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.