Arithmetic Progression

To recall, progression is a sequence of numbers in a particular order. If we observe in our regular lives, we come across progression quite often. Roll numbers of a class, days in a week or months in a year. Did you notice that counting numbers, even numbers or odd numbers, all follow a particular progression! To understand this let us study the topic of arithmetic progression in detail.

Table of Contents:

Arithmetic Progression (AP)

What is Arithmetic Progression in Maths?

Arithmetic progressions is a mathematical sequence in which the difference between two consecutive terms is always a constant.

A progression is a special type of sequence for which it is possible to obtain a formula for the nth term. The arithmetic progression is the most commonly used sequence in maths with easy to understand formulas.

Formal Definitions:

We will refer arithmetic progression with the abbreviation AP from now on.

Definition 1: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.

Definition 2: The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.

Let’s understand the concepts of AP:

Consider an AP to be:

a1, a2, a3, ……………., an

So, a2 – a1 = a3 – a2 = ……. = an – an – 1 = d,

Where, “d” is the common difference.

The progression can also be written in terms of common difference, as follows

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d.

Let  “a” is the first term of the progression and “d” is the common difference, then

Position of Terms Representation of Terms Values of Term
1 a1 a + d = a + (1-1) + d
2 a2 a + 2d = a + (2-1) + d
3 a3 a + 3d = a + (3-1) + d
4 a4 a + 4d = a + (4-1) + d
. . .
. . .
. . .
. . .
 n an a + (n-1)d

Notation:

Common terms used in arithmetic progression (AP) are denoted as:

d = Common difference

a = Fist term

l = last term

an = nth term

Sn = Sum of the first n terms

nth Term of an Arithmetic Progression:

For a given AP, where “a” is the first term, “d” is the common difference, “n” is the number of terms in an AP and “an” be the last term, the relation is given as

an = a + (n − 1) × d

Sum of First “n” Terms of an Arithmetic Progression

For any progression, the sum of n terms can be easily calculated. For an AP, the sum of the first n terms can be calculated if the first term and the total terms are known. This formula is explained below:

Consider an AP consisting “n” terms.

Sum of n term is given as:

S = n/2[2a + (n − 1) × d]

Proof:

Consider an AP: a, a + d, a + 2d, …………., a + (n – 1) × d

Sum of first n terms = a + (a + d) + (a + 2d) + ………. + [a + (n – 1) × d] ——————-(i)

Writing the terms in reverse order,

we have: S = [a + (n – 1) × d] + [a + (n – 2) × d] + [a + (n – 3) × d] + ……. (a) ———–(ii)

Adding both the equations term wise, we have

2S = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + …………. + [2a + (n – 1) ×d] (n-terms)

2S = n × [2a + (n – 1) × d]

S = n/2[2a + (n − 1) × d]

Examples

Example 1: Find the value of n. If a = 10, d = 5, an = 95.

Solution:

Given, a = 10, d = 5, an = 95

From the formula of general term, we have:

an = a + (n − 1) × d

95 = 10 + (n − 1) × 5

(n − 1) = 17

n = 18

Example 2: Find the 20th term for the given AP:

3, 5, 7, 9, ……

Solution:

Given, a = 3, d = 5 – 3 = 2, n = 20

an = a + (n − 1) × d

an = 3 + (20 − 1) × 2

an = 3 + 38

⇒an = 41

Example 3: Find the sum of first 30 multiples of 4.

Solution: Given,

a = 4, n = 30, d = 4

We know,

S = n/2 [2a + (n − 1) × d]

S = 30/2[2 (4) + (30 − 1) × 4]

S = 15[8 + 116]

S = 1860

Practice Questions

Question 1: Find the a_n and 10th term of the progression: 3,10,17,24.

Question 2: If a = 2, d = 3 and n = 90. Find an  and Sn.

Question 3: The 7th term and 10th terms of an AP is 12 and 25. Find the 12th term.

To learn more about arithmetic progression in an interactive way, Download  BYJU’S-The Learning App.

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Practise This Question

If 9th term of an AP is 15 and the common difference is 2, then the 1st term of the AP is