In this blog, we will learn about Arithmetic Progressions.

Arithmetic Progression is a sequence of numbers that has a constant difference between every two consecutive terms.

In other words, Arithmetic Progression is a list of numbers in which each term except the first term is the result of adding the same number, called the common difference, to the preceding term.

Common difference of the AP

Let us take an example of an Arithmetic Sequence. The sequence 5, 11, 17, 23, 29, 35 . . . is an arithmetic sequence; Because the same number, 6 is added to each term of the sequence to get the succeeding term.

This fixed number is called the common difference of the AP. Remember that common differences can be positive, negative, or zero.

Let us denote,

The first term of an AP = a₁,

Second term = a₂ ...

n^th term = a_n and

The common difference = d.

Then the AP becomes,

a₁, a₂, a₃ ... a_n So, a₂ � a₁ = a₃ � a₂ = ... = a_n � a_{n � 1} = d

a, a + d, a + 2d, a + 3d, ...

Represents an arithmetic progression where a is the first term and d is the common difference. This is called the general form of an AP.

Types Of Arithmetic Progressions

Arithmetic progressions (A.P.) can be of 2 types:

� Finite A.P. � This type of Arithmetic Progressions (A.P.) has the last term. And the number of terms in an AP is fixed.

For example, The cash prizes (in Rs.) given by a school to the toppers of Classes I to X are, respectively, 200, 250, 300, 350 ... 750.

� Infinite A.P. � These APs do not have the last term. And the number of terms is not fixed, that is why it is called Infinite A.P.

For example, 100, 70, 40, 10 ... In this A.P., here is not the last term.

Now, to know about an AP, what is the minimum information that you need? Is it enough to know the first term? Or, is it enough to know only the common difference? You will find that you will need to know both � the first term a and the common difference d.

If the first term a = 5 and the common difference d = 2, then the AP is 5, 7, 9, 11 ...

And if a = 5 and d = � 2, then the AP is 5, 3, 1, �1 ...

So, if you know what a and d are, you can list the AP. Now, If you are given a list of numbers can you say that it is an AP and then find a and d?

For example, for the list of numbers: 6, 9, 12, 15, ...

Since a is the first term, it can easily be written.

Here a₁ = 6, a₂ = 9, a₃ = 12 and a₄ = 15

We know that in an AP, every succeeding term is obtained by adding d to the preceding term. So, d found by subtracting any term from its succeeding term, i.e., the term which immediately follows it should be the same for an AP.

We have,

a₂ � a₁ = 9 � 6 = 3,

a₃ � a₂ = 12 � 9 = 3,

a₄ � a₃ = 15 � 12 = 3

Here the difference between any two consecutive terms in each case is 3. So, the given list is an AP whose first term a is 6 and the common difference d is 3.

In general,

For an AP a₁, a₂, a₃ ... a_n, we have

d = a_{k + 1} � a_k

Where a_{k + 1} and a_k are the (k + 1)^th and the k^th terms respectively.

To obtain d in a given AP, we need not find all of a₂ � a₁, a₃ � a₂, a₄ � a₃ ... It is enough to find only one of them.

Read More:
Nth term of an AP (Arithmetic Progression):�Formula and Sum of n Terms