Common Difference

The term “common difference” is a concept that is used in sequences and arithmetic progressions. A common difference is a difference between any term and its preceding term in an arithmetic sequence, and it is generally represented by the letter “d”. So, to determine the common difference of an arithmetic sequence, subtract the first term from the second term, the second term from the third term, etc. The common difference of an arithmetic sequence plays a vital role in determining the successive terms of the sequence. 

For instance, 3, 7, 11, 15, 19, 23,… is an arithmetic sequence. The common difference between any two successive terms of a given sequence is 4, i.e.,

1. Second term – First term = 7 – 3 = 4
2. Third term – Second term = 11 – 7 = 4
3. Fourth term – Third term = 15 – 11 = 4
4. Fifth term – Fourth term = 19 – 15 = 4, and so on.

There are two types of arithmetic progression based on the common difference, i.e., 

An arithmetic progression increases if the common difference is positive, and an arithmetic progression decreases if the common difference is negative. 

For example, 2, 4, 6, 8, 10, 12,… is an increasing arithmetic progression as the common difference is positive, i.e., “2”. 10, 4, -2, -8, -14,… is a decreasing arithmetic progression as the common difference is negative, i.e., “-6”.

Note: Common difference of an arithmetic sequence remains the same if a constant quantity is added or subtracted to or from each term of the arithmetic sequence. 

For example, 4, 9, 14, 19, 24, 29,… is an arithmetic progression with a common difference of “5”. If “7” is added to each term of the given progression, then it becomes 11, 16, 21, 26, 31, 26,… and the common difference between the successive terms of the new sequence is also “5”. If “2” is subtracted from each term of the given progression, then it becomes 2, 7, 12, 17, 22, 27,… and the common difference between the successive terms of the new sequence is also “5”.

How to find the common difference of an Arithmetic Progression? A sequence is also known as a progression and is defined as the successive arrangement of numbers in order while following some specific rules. Depending upon the set of rules followed by a sequence, it is classified into various kinds, such as an arithmetic sequence, geometric sequence, harmonic sequence, and Fibonacci sequence. An arithmetic sequence or progression is a sequence of numbers when the difference between any two successive numbers is the same. For instance, 5, 10, 15, 20, 25, 30,… is an arithmetic sequence, where the common difference between any two successive terms is 5.

General Form of an AP:

AP = a, a + d, a + 2d, a + 3d,……., a + (n-1) d

Where,

“a” is the first term of the progression, and

“d” is a common difference.