Nth term of an Arithmetic Progression

The nth term of an arithmetic progression (AP) is an important concept in mathematics that helps find any term of a sequence without writing all the previous terms. It is based on a constant pattern where each term increases or decreases by a fixed difference. In this guide, you will learn about the nth term of an arithmetic progression, its formulas with easy steps, and examples. 

Table of Contents:

• What is an Arithmetic Progression
• What is the nth term of an Arithmetic Progression
• How to Find the nth term of an Arithmetic Progression
• Solved Examples on nth term of an Arithmetic Progression

What is an Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. That fixed difference is called the common difference, written as d.

In general notation, an AP is written as a, a+d, a+2d, a+3d, … where a is the first term and d is the common difference. Both a and d can be positive, negative, or even zero, but d remains constant throughout.

Read more: Important Questions on Arithmetic Progression - Class 10

What is the nth term of an Arithmetic Progression

The nth term of an AP is the term in the nth position in the sequence. It's also called the general term of an AP, and it's usually written as   an

• Derivation of the nth term of an Arithmetic Progression

Let  a1,a2,a3,...be an AP whose first term  a1is a and the common difference is d.

 

The coefficient of d in each term is always one less than the position number.
∴ the general term of an AP is  an= a + (n - 1)d
Where 'a' = first term, 'd' = common difference, and 'n' = position of the term

How to Find the  nth term of an Arithmetic Progression 

Here are the four steps to follow to find the nth term of an arithmetic progression:

•  Identify the first term (a).

• Find the common difference (d). Subtract the first term from the second: d = a2−a1

• Note the value of n (which term you need). For 'find the 15th term', n = 15. 

• Substitute into  an = a + (n − 1)d and simplify.
  
  

Solved Examples on the nth term of an Arithmetic Progression 

Example 1: Find the 12th term of the AP: 5, 8, 11, 14, …

Solution: Given:  a = 5,   d = 8 − 5 = 3,   n = 12

 an = a + (n − 1)d

 a12 = 5 + (12 − 1) × 3

 a12 = 5 + 11 × 3 = 5 + 33 = 38

The 12th term of the AP is 38.

Example 2: How many terms are in the AP: 7, 13, 19, 25, ..., 205?

Solution: Given, a = 7,   d = 13 − 7 = 6, and an= 205

205 = 7 + (n − 1) × 6

205 − 7 = (n − 1) × 6

198 = (n − 1) × 6

n − 1 = 33  

∴n = 34

There are 34 terms in the AP 7, 13, 19, 25, ..., 205

Example 3: Check whether 301 is a term of the AP: 5, 11, 17, 23, …

Solution: Given a = 5,   d = 6

Assume 301 =  an = 5 + (n − 1) × 6

301 − 5 = (n − 1) × 6

296 = (n − 1) × 6

n − 1 = 296/6 = 49.33…  (not an integer)

Since n is the position of the term, it must always be a positive integer. Hence, 301 is not a part of the arithmetic progression.

Example 4: The 4th term of an AP is 0, and the 10th term is −12. Find the AP.

Solution:  a4= a + 3d = 0  … (1)

 a10 = a + 9d = −12  … (2)

Subtracting (1) from (2):

6d = −12  

d = −2

Substituting d = −2 in (1) a + 3(−2) = 0 ⇒ a = 6

a = 6 and d = -2

∴ AP is  6, 4, 2, 0, −2, −4, −6, −8, −10, −12, …