What is the Common Difference in an Arithmetic Sequence

Arithmetic progressions are the sequences in which the terms continue to increase or decrease with a steady value. The difference that is observed in those terms is called the common difference for that progression. Common differences can be used to find the factors of a progression that are not in sight just yet. Using this we can calculate the nth term of an A.P., and the sum of that A.P. Hence, it won’t be wrong to say that it is the most important constituent of an A.P.

The common difference between the two terms and the nth term

The common difference for an arithmetic progression is the entity that continues to remain constant in a process no matter which two consecutive terms you are choosing from an arithmetic progression. Besides, it won’t be wrong that an arithmetic sequence is only a function of the common difference for that particular sequence. 

If the common difference for a sequence is known, then it is possible to extend a given sequence to as many terms as you want. The common difference is the difference between any two consecutive terms of a progression.

The common difference for an A.P. can be given by the following formula:

d = a_{n + 1} – a_n

This value remains constant for any pair of consecutive terms that are chosen from a progression. This is an entity that is much needed to calculate the nth term of an A.P. as well the sum of that progression. 

Even if only the first and nth term of an A.P. is known to us, it is possible to calculate the common difference by using the common principle of transposition in the following formula:

a_n = a + ( n – 1 ) × d

And, this is the exact formula that is used to represent the nth term of an arithmetic progression. This term can be found using the first term and the common difference for an A.P.

Common difference formula

The common difference in an arithmetic progression can be found by using the two consecutive terms of an A.P. To get the common difference out of those terms, you will need to subtract the first term from the second one. That will give you the account of common differences for that sequence. 

Let a_n and a_{n + 1} be the two consecutive terms of an A.P.

Then, 

The common difference for that sequence can be given in the following way:

d = a_{n + 1} – a_n

Also, it can be represented in the way of:

d = a_n – a_{n – 1}

It is also possible to find the common difference of an A.P. if its first and nth terms are given. To find the common difference that way, you will need to put the respective values in the formula for the nth term of an A.P. Using that formula, you will be able to find out the common difference for that progression quite easily. 

On the other hand, if only the sum, first term and the last term of an A.P. are given, you will need to apply the formula of the sum of a sequence to get the number of terms of the A.P. beforehand. Then, you can use the formula for the nth term of an A.P.

Common differences can be positive, negative or zero

The value for the common difference of an arithmetic progression can be positive, negative as well as zero. However, it is to be noted that the value of common difference will have a considerable effect on that A.P.

Now, 

When d = 0

Let us assume am A.P for which the common difference is zero.

The general form of an arithmetic progression is as follows:

a, a + d, a + 2d, a + 3d, ….

As the value of d = 0

The modified arithmetic progression is:

a, a, a, a, ….

Also, it can be seen that the common difference between any two terms of this A.P. is always zero. Hence, it can be said that it is possible to make an A.P. whose common difference is zero. However, all the terms of that A.P. will be ‘a’ or the same as the first term. 

When d = positive

Take the progressions 2, 6, 10, 14, …. For example,

The common difference for the A.P. is 4, which is a positive integer. Hence it can be concluded that the common difference between an A.P. can be positive.

When d = negative

Let there be a decreasing A.P.

30, 27, 24, 21, ….

In this progression, the common difference is -3.

It can be seen that the result will remain the same for any pair of consecutive terms chosen. Hence, it is possible to make an A.P. whose common difference is negative.

Conclusion

Common differences alongside the first term happen to be the most important factors that are needed to form an arithmetic progression. By utilising both of these, it is possible to extend a progression to as many terms as you want to. There are several ways in which the common differences can be calculated. It can be made as complicated as possible, but it is just as much simpler if only two consecutive terms of the A.P. are given.