Arithmetic Sequence – Formula, Meaning, Examples

Sequence and series are crucial topics in mathematics. A sequence is a  list of elements in which repetitions of any sort are allowed, whereas the sum of all elements is known as a series. For example, 2, 4, 6, 8, 10 is a sequence with five elements and the corresponding series will be 2+4+6+8+10, where the sum of the series will be 30. Now let us understand them.  An AP(arithmetic progression) is one example of sequence and series.

Sequence

It is defined as the collection of numbers which are arranged in a specific pattern where each number in the sequence is known as a term.

For example- 5, 10, 15, 20, 25, … is a sequence where the ellipsis sign at the end of the sequence represents that the list continues further till infinity. In this sequence 5 is the first term, 10 is the second term,15 is the third term and so on. Each component in the sequence has a common difference, and the order continues with the common difference. In the example given above, the common difference is 5. The types of sequences present are:

• Arithmetic Sequence

• Geometric Sequence

• harmonic Sequence

• Fibonacci Sequence

Arithmetic Sequence

It is a sequence in which every term is found by adding or subtracting a definite number(common difference) to the preceding number.

x₁,  x₁+ m , x₁+ 2m, x₁ + 3m,….. x₁ + (n-1)m and so on, where m is the common difference.

For example- 1, 4, 7, 10,13,16…….

Here, we observe that the common difference is 3.

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}

The sum of the first ‘n’ terms of AP

The formula is

S_n = ( n / 2 ) [ 2a + ( n – 1) d ]

Now, let us try to prove this formula true for the first term of an Arithmetic progression.

Finding the sum of only the first term of the given A.P.

S₁ = ( 1 / 2 ) [ 2a + ( 1 – 1) d ]

S₁ = ( 1 / 2 ) [ 2a ]; 0d = 0;

S₁ = a

Hence, it can be said that the formula stands for all the terms of an A.P.

This formula for the sum of an Arithmetic progression can be used only when the nth term for that A.P. is not known to us. In that case, the formula for the sum of n terms of an A.P. can be stated in the following way:

S_n = ( n / 2 ) [ a₁ + a_n ]

Common differences can be +ve, -ve 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. It is the first term of an arithmetic progression that initiates a progression. By using this value we can compute any term of that sequence if only the common difference for that progression is known to us. By using the significant values in the formula for the nth term for the A.P. you can find any term that you want. It is also possible to find the sum for n terms of that A.P., and the whole process becomes much more effective if the last term of the A.P. is also known.