Arithmetic Progression

Arithmetic progression (AP) is a series of numbers obtained by adding a constant value to the preceding term, also known as an arithmetic sequence. For example, the series of all odd natural numbers 1, 3, 5, 7, 9…. is an AP, where 1 is the first term of the series. This is because 1 is the first natural odd number, and adding the constant value 2 to the first term will give us our next odd number: 3. Similarly, adding 2 again to the second term—3—will give the third term: 5. Repeating this again and again will give us a series of all natural odd numbers. 

This can also help obtain other series, such as the series of all natural numbers, series of all even natural numbers etc. 

As we know, mathematics is everywhere in our daily lives. Let’s consider some examples of AP in our daily lives. We can use AP to find:

● The number of students with odd or even roll numbers

● The number of months with 31 days

● The number of Sundays in a year

Main terms and their notations

● Common difference, d

● First term of an AP, a1

● nth term of an AP, a(n)

● Last term of an AP, L

● Sum of the first n terms, Sn

These are the common terms used to represent an AP. You will also find these terms in the common formulas of an AP.

Introduction to common AP terms

Let’s take a sample AP to understand the important terms.

Series of all even natural numbers: 2, 4, 6, 8….. till infinity

Common difference in an AP, d

Common difference is the difference between two consecutive numbers in an AP. Moreover, we get the second AP term by adding common differences to the first term. Repeating this with every preceding term will give us the next term. Common differences in an AP can have a denotation “d”. To find the common difference of an AP, you can subtract the first term from the second term. You can also subtract any term from its preceding term to get the value of common difference. 

In our sample series of all even natural numbers: 2, 4, 6, 8, ….. 

Common difference (d) = a2 – a1 

Therefore, d = 4 – 2 = 2 

To verify whether the common difference is true for all the terms in an AP, 

a2 – a1 = a4 – a3

4 – 2 = 8 – 6

2 = 2

First term of an AP, a¹

As simple as its name, the first term of an AP is the first unit of the series. The term can have a denotation “a¹”. In our sample series of all even natural numbers, 2 is the first term for an obvious reason: 2 is the first even number in the series of natural numbers.

An AP can be created through the first term and common difference of an AP, 

a, a + d, a + 2d, a + 3d…….

nth term of an AP, a(n)

The nth term is the term around which the calculation has to be done. It can be any natural number.

Formulas to solve AP questions

The general form of an AP

 This formula can help create an AP from the first term “a” and common difference “d”. 

AP = a, a + d, a + 2d, a + 3d,……

Formula to find the nth term of an AP 

This formula can help find the nth term of an AP, where “n” can be any natural number. Moreover, you can find the first term “a” or common difference “d” of an AP through this formula if the nth term is present. 

a(n) = a + (n-1) × d

Sum of n terms in an AP 

This formula can help find the sum of the first n terms of an AP. Similar to the nth term formula, you can find the first term or common difference through this formula. Moreover, you can find nth term if the value of Sn is present. 

Sn = n/2 [2a + (n-1) × d] 

Sum of all terms of an AP

To find the sum of all terms present in a finite AP,

Sum of all terms = n/2 ( a+ l) 

Solved AP questions

Q. Find the 12th term of AP: 1, 5, 9, 13, 17,…..

 Ans. To find the 12th term of this AP,

 a(n) = a + (n-1) × d

 n = 12a12 = 1 + (12-1) × 4

a12 = 1 + 44

a12 = 45 

Q. If a = 4, d = 3, and n = 90, find a(n) and Sn.

 To find an,

 an = a + (n-1) × d

 n = 90

 an = 4 + (90-1) × 3

an = 4 + 267

an = 271

 To find Sn,

 Sn = n/2 [2a + (n-1) × d]

 Sn = 45 [8 + (90-1) × 3]

Sn = 45 [8 + 267]

Sn = 12195

Conclusion

AP has a good weightage in the competitive exams. While this was a brief introduction, the importance of AP is clear in everyday life.We proved various formulas related to AP like nth term of an AP and sum of n terms in AP. We also took some examples to strengthen our concepts.