Before knowing about what arithmetic progression is, let us learn about some basic terminologies that will help us in understanding the chapter.

Sequences: It is an arrangement of numbers in definite order or specific pattern that forms a sequence. 

e.g. i) 2, 4, 6, 8, …..

      ii) 3, 6, 12, 24, 48, …..

     iii) 1, 4, 9, 16, 25, …..

In general, the nth term of a sequence is denoted by an or tn and sequence is denoted by < an > or { an }.

Types of Sequences

1. Arithmetic Progression (AP)

2. Geometric Progression (GP)

3. Harmonic Progression (HP)

In this chapter, we will learn about Arithmetic Progression (AP).

Arithmetic Progression

• An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

• The first term of an AP is denoted by ‘a’.

• This fixed number is called the common difference of the AP and is denoted by ‘d’. 

• Each number which forms an AP is called the term of an AP.

Note: 

1. Common difference can be positive, negative or zero.

2. Second term - first term = third term - second term = d (common difference)                             

                 (a2 - a1)                          (a3 - a2)

e.g. i) 1, 2, 3, 4, …..

      ii) 100, 70, 40, 10, …..

      iii) 3, 3, 3, 3, …..

Let us now go through some examples and check whether the given series is an AP or not?

Q. Which of the following is an AP? If they form an AP, find the common difference d?

i) 2, 4, 8, 16

Sol. first term, a = 2

Common difference, d = a2 - a1

                                     = 4 - 2

                                     = 2

Also, a3 - a2 = 8 - 4

                     = 4

Since, common differences are not equal.

Hence, given series is not an AP.

ii) -10, -6, -2, 2

Sol. first term, a = -10

Common difference, d = a2 - a1

                                     = -6 - (-10)

                                     = 4

Also, a3 - a2 = -2 - (-6)

                     = 4

Since, common differences are equal.

Hence, given series is not an AP.

Alright! Now you have learnt about how to identify whether a given series is AP or not,  let us move to the next step in the chapter and play with some numericals.

nth Term: The term which represents any term of a given arithmetic progression is known as the ‘nth term’ of the AP.

If ‘a’ is the first term and ‘d’ is the common difference of an AP then the nth term of AP is given by:

nth term(an) = a + (n-1)d

where,  n = number of terms

             a = first term

             d = common difference

Q. Find the 8th term of an Ap whose first term is 7 and the common difference is 3.

Sol. Given: a = 7, d = 3

       We have to find the 8th term i.e, a8 

Using the formula for the nth term of an AP for n = 8.

an = a + (n-1)d

For n = 8

     a8 = 7 + (8-1)3

  a8 = 7 + 21

  a8 = 28

Hence, the 8th term of the given AP is 28.

Now that we have a basic idea about the nth term of an AP. To grasp this more efficiently, let’s do one more example.

Q. Naveen saved ₹5 in the first week of a year and then increased her weekly savings by ₹1.75. If in the nth week, her total savings become ₹20.75, find n. 

Sol. According to given conditions,

       a =5, d=1.75, an=20.75, n=?

       We know that,

             an = a + (n-1)d

         20.75 = 5 + (n-1)1.75

         15.75 = (n-1)1.75

         n-1 = 9

         n = 10

Hence, the value of n is 10.

Sum of First ‘n’ terms of an AP

If ‘a’ and ‘d’ are the first term and common difference of an AP respectively, then the sum of first n terms of an AP is denoted by Sn and given by:

Sn = n2[2a + (n-1)d]

Let us do a few examples to understand and memorize this formula further.

Q. Find the sum of the following AP’s.

i) 2, 7, 12, ……… to 10 terms

Here, a=2, d=7-2 =5, n =10