The nth term of an AP is given by aₙ = a + (n − 1)d, where a is the first term, d is the common difference, and n is the term number. For example, in the AP 2, 7, 12, …, the 10th term = 2 + (10−1)×5 = 47.

Q: Can the common difference in an AP be negative or zero?

Yes. If d > 0, the AP is increasing (like 2, 5, 8, …). If d = 0, all terms are equal (like 7, 7, 7, …). If d < 0, the AP is decreasing (like 20, 15, 10, 5, …).

Q: How do I check if three numbers a, b, c are in AP?

Simply check if 2b = a + c. If this condition holds, then a, b, c are in AP, and b is the arithmetic mean of a and c.

Q: What is arithmetic mean and how is it related to AP?

The Arithmetic Mean (AM) of two numbers a and b is (a+b)/2. In an AP, every middle term is the AM of the two terms surrounding it. If three numbers are in AP, the middle number is always the AM of the other two.

Q: What is the sum of first n natural numbers?

The sum of the first n natural numbers (1 + 2 + 3 + … + n) is given by n(n+1)/2.

Chapter 5: Arithmetic Progression Notes for Class 10 are available in this article. This clear, student-friendly guide breaks down arithmetic progressions (APs) — from identifying the first term and common difference to finding the nth term and the sum of the first n terms using simple formulas. Aligned with the NCERT syllabus and CBSE exam patterns, it explains how to spot whether a sequence is an AP, when to use a_n and S_n , and how to solve word problems involving growth, savings, seating arrangements and instalments. You’ll find step-by-step worked examples, shortcut tips for quick calculations of n, a, d and S_n, and exam-smart strategies for checking answers and avoiding common sign errors. These notes include quick revision bullets, practice problems with answers, and assertion–reason questions to build confidence before tests..

Chapter 5: Arithmetic Progression Notes for Class 10

1. What is an Arithmetic Progression?

Definition of Arithmetic Progression

An Arithmetic Progression (AP) is a list of numbers in which each term after the first is obtained by adding a fixed number to the previous term. This fixed number is called the common difference.

General Form of an AP:

a, a+d, a+2d, a+3d, …

where

a = first term
d = common difference

How to Find the Common Difference

Common difference = (any term) − (the term before it)

d = aₙ − aₙ₋₁

Example 1: Is 3, 7, 11, 15, 19, … an AP? If yes, find the common difference.

Solution: 7 − 3 = 4, 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4

The difference between every pair of consecutive terms is constant = 4.

Yes, it is an AP. Common difference d = 4

Finite and Infinite AP

Type	Description	Example
Finite AP	Has a definite number of terms (has a last term).	2, 5, 8, 11, 14 (5 terms)
Infinite AP	Continues endlessly (has no last term).	2, 5, 8, 11, 14, 17, …

2. nth Term (General Term) of an AP

nth Term Formula

aₙ = a + (n − 1) × d

where

aₙ = nth term
a = first term
n = term number
d = common difference

Where Does This Formula Come From?

Let's think about it logically. In an AP starting at a:

1st term = a (add d zero times)

2nd term = a + d (add d once)

3rd term = a + 2d (add d twice)

$%5Cvdots$

nth term = a + (n−1)d (add d exactly n−1 times)

Example 1: Find the 10th term of the AP: 2, 7, 12, 17, …

a = 2, d = 7 − 2 = 5, n = 10

a₁₀ = 2 + (10 − 1) × 5 = 2 + 9 × 5 = 2 + 45

a₁₀ = 47

Example 2: Which term of the AP: 3, 8, 13, 18, … is equal to 78?

a = 3, d = 5. Set aₙ = 78

78 = 3 + (n − 1) × 5

⇒ 75 = (n − 1) × 5 ⇒ n − 1 = 15 ⇒ n = 16

78 is the 16th term of the AP.

3. Types of AP & Behaviour of Common Difference

Common Difference	Type	Behaviour	Example
d > 0	Increasing AP	Each term is bigger than the last.	2, 6, 10, 14, … (d = 4)
d = 0	Constant AP	All terms are equal.	5, 5, 5, 5, … (d = 0)
d < 0	Decreasing AP	Each term is smaller than the last.	20, 15, 10, 5, … (d = −5)

4. Sum of n Terms of an AP

The Derivation

Sₙ = a + (a+d) + (a+2d) + … + (l−d) + l …(1)

Sₙ = l + (l−d) + (l−2d) + … + (a+d) + a …(2)

Adding (1) and (2): 2Sₙ = n × (a + l)

Since l = a + (n−1)d, substituting gives: Sₙ = n/2 × [2a + (n−1)d]

Sum of First n Terms: when a and d are known

Sₙ = n/2 × [2a + (n − 1)d]

Use when first term (a) and common difference (d) are given

Sum of First n Terms: when first and last term are known

Sₙ = n/2 × (a + l)

l = last term = a + (n−1)d

Use when you know both ends of the AP

Example:

Find the sum of the first 22 terms of the AP: 8, 3, −2, …

a = 8, d = 3 − 8 = −5, n = 22

⇒ S₂₂ = 22/2 × [2×8 + (22−1)×(−5)] = 11 × [16 + (21)(−5)]

⇒ = 11 × [16 − 105] = 11 × (−89)

S₂₂ = −979

Important Relationship: aₙ = Sₙ − Sₙ₋₁
The nth term = Sum of n terms − Sum of (n−1) terms

aₙ = Sₙ − Sₙ₋₁ (valid for n ≥ 2)

Also note: a₁ = S₁

Sum of First n Natural Numbers

The natural numbers 1, 2, 3, …, n form an AP with a = 1 and d = 1. So:

Sum of First n Natural Numbers

Sₙ = n(n + 1) / 2

Example: Sum of 1 to 10 = 10 × 11 / 2 = 55

5. Arithmetic Mean

Arithmetic Mean of Two Numbers

A.M. = (a + b) / 2

If a, A, b are in AP, then A is the arithmetic mean of a and b

If three numbers a, b, c are in AP, then b = (a + c) / 2, which means 2b = a + c.
Example 1: If x, 12, y are in AP, and x + y = 24, find x and y.

Solution: Since x, 12, y are in AP: 2 × 12 = x + y

⇒ x + y = 24

We need another condition. Given x + y = 24, and 12 is the AM of x and y.

⇒ Any pair with sum 24 works, e.g., x = 10, y = 14 with d = 2 or x = 8, y = 16 with d = 4.

6. All Formulas at a Glance

Concept	Formula
First term	a
Common difference	$d%20%3D%20a_n%20-%20a_%7Bn-1%7D$
General form	a, a+d, a+2d, a+3d,...
nth term (general term)	$a_n%20%3D%20a%20%2B%20(n-1)d$
Last term ((l))	l = a + (n-1)d
Sum Formula 1 (using d)	$S_n%20%3D%20%5Cfrac%7Bn%7D%7B2%7D%5B2a%20%2B%20(n-1)d%5D$
Sum Formula 2 (using l)	$S_n%20%3D%20%5Cfrac%7Bn%7D%7B2%7D(a%20%2B%20l)$
nth term from (S_n)	$a_n%20%3D%20S_n%20-%20S_%7Bn-1%7D$
Arithmetic Mean of (a) and (b)	$AM%20%3D%20%5Cfrac%7Ba%2Bb%7D%7B2%7D$
Sum of first (n) natural numbers	$%5Cfrac%7Bn(n%2B1)%7D%7B2%7D$
If (a), (b), (c) are in AP	2b = a + c

Click below to download your free Class 10 Maths Chapter 5: Arithmetic Progression PDF Notes perfect for last-minute CBSE board exam revision.

Class 10 Maths Chapter 5: Arithmetic Progression PDF Notes

Frequently Asked Questions of Chapter 5: Arithmetic Progression Notes for Class 10

1. What is the formula for the nth term of an AP?

The nth term of an AP is given by aₙ = a + (n − 1)d, where a is the first term, d is the common difference, and n is the term number. For example, in the AP 2, 7, 12, …, the 10th term = 2 + (10−1)×5 = 47.

2. Can the common difference in an AP be negative or zero?