CBSE Class 10 Notes | Arithmetic Progressions Chapter 5
CBSE Class 10 Maths Notes – Chapter 5 Arithmetic Progressions
Welcome to the most comprehensive CBSE Class 10 Notes for Mathematics Chapter 5: Arithmetic Progressions. These NCERT-aligned Class 10 Maths Chapter 5 Notes are designed specifically for CBSE Board Exam 2025-26 preparation, covering every concept, formula, and question type you need to score full marks.
Arithmetic Progressions (AP) is one of the most scoring chapters in CBSE Class 10 Mathematics, carrying 4-5 marks in the board examination. Whether you're preparing for Maths Standard or Basic, these AP Class 10 Notes will help you master sequences, nth term formulas, and sum calculations with ease.
Table of Contents
1. Introduction to Arithmetic Progressions
What is an Arithmetic Progression (AP)?
An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).
Examples of AP:
- 2, 5, 8, 11, 14... (d = 3)
- 10, 7, 4, 1, -2... (d = -3)
- 5, 5, 5, 5... (d = 0)
Key Components of AP
| Component | Symbol | Description |
|---|---|---|
| First Term | a | The initial term of the sequence |
| Common Difference | d | Fixed difference between consecutive terms |
| nth Term | aₙ | The term at position n |
| Number of Terms | n | Total count of terms in the AP |
| Last Term | l | The final term of a finite AP |
CBSE Board Exam Importance
📊 Chapter Weightage: 4-5 Marks (Both Standard & Basic)
📝 Question Types: MCQs (1 mark), Short Answer (2-3 marks), Long Answer (5 marks)
🎯 NCERT Alignment: Strictly follows NCERT Book Chapter 5 (Exercises 5.1 to 5.4)
📝 Question Types: MCQs (1 mark), Short Answer (2-3 marks), Long Answer (5 marks)
🎯 NCERT Alignment: Strictly follows NCERT Book Chapter 5 (Exercises 5.1 to 5.4)
2. Chapter Overview: Arithmetic Progressions Class 10
Topics Covered in CBSE Class 10 Notes
- Introduction to Sequences and Series
- Definition and Examples of AP
- General Form of Arithmetic Progression
- nth Term of an AP (aₙ = a + (n-1)d)
- Sum of First n Terms of an AP
- Properties of Arithmetic Progressions
- Arithmetic Mean between Two Numbers
- Word Problems and Real-Life Applications
Types of Board Questions You Can Expect
| Question Type | Marks | Frequency |
|---|---|---|
| Identifying AP sequences | 1 Mark | Very High |
| Finding nth term | 2-3 Marks | High |
| Calculating sum of n terms | 3-5 Marks | Very High |
| Word problems | 4-5 Marks | High |
| Case Study Questions | 4 Marks | Moderate |
3. Key Concepts: NCERT-Based Explanations
📌 Definition of Arithmetic Progression
A sequence a₁, a₂, a₃, ..., aₙ is called an Arithmetic Progression if:
a₂ - a₁ = a₃ - a₂ = a₄ - a₃ = ... = aₙ - aₙ₋₁ = d (constant)
Finite AP: Has limited number of terms (e.g., 2, 4, 6, 8, 10)
Infinite AP: Continues indefinitely (e.g., 3, 6, 9, 12, ...)
📌 General Form of AP
Standard Form: a, a + d, a + 2d, a + 3d, ..., a + (n-1)d
📌 nth Term Formula (Most Important)
nth Term Formula
aₙ = a + (n - 1)d
Where: aₙ = nth term | a = First term | n = Position | d = Common difference
Example: Find the 15th term of AP: 3, 7, 11, 15...
Solution: a = 3, d = 4, n = 15
a₁₅ = 3 + (15-1)×4 = 3 + 56 = 59
Solution: a = 3, d = 4, n = 15
a₁₅ = 3 + (15-1)×4 = 3 + 56 = 59
📌 Sum of First n Terms Formula
Formula 1 (When last term unknown)
Sₙ = n/2 [2a + (n - 1)d]
Formula 2 (When last term l known)
Sₙ = n/2 (a + l)
📌 Finding the Middle Term
For an AP with odd number of terms (n), the middle term is the ((n+1)/2)th term.
Example: In AP with 11 terms, middle term = 6th term
4. Important Formulas Box
Must-Know Formulas for CBSE Class 10 Board Exams
| Formula | Expression | When to Use |
|---|---|---|
| nth Term | aₙ = a + (n - 1)d | Finding any specific term |
| Sum of n Terms (V1) | Sₙ = n/2 [2a + (n - 1)d] | When you know a, d, and n |
| Sum of n Terms (V2) | Sₙ = n/2 (a + l) | When first and last term known |
| Common Difference | d = a₂ - a₁ | To verify if sequence is AP |
| Number of Terms | n = [(l - a)/d] + 1 | Finding total terms |
| Arithmetic Mean | AM = (a + b)/2 | Single mean between two numbers |
5. Solved Examples (CBSE Board Pattern)
Example 1: Easy (Finding nth Term) - 2 Marks
Question: Find the 20th term of the AP: 5, 9, 13, 17...
Step 1: Identify values
a = 5, d = 9 - 5 = 4, n = 20
Step 2: Apply formula
aₙ = a + (n - 1)d
Step 3: Substitute
a₂₀ = 5 + (20 - 1) × 4
a₂₀ = 5 + 19 × 4
a₂₀ = 5 + 76
Answer: a₂₀ = 81
a = 5, d = 9 - 5 = 4, n = 20
Step 2: Apply formula
aₙ = a + (n - 1)d
Step 3: Substitute
a₂₀ = 5 + (20 - 1) × 4
a₂₀ = 5 + 19 × 4
a₂₀ = 5 + 76
Answer: a₂₀ = 81
Example 2: Moderate (Finding Sum) - 3 Marks
Question: Find the sum of first 25 terms of AP: 3, 7, 11, 15...
Given: a = 3, d = 4, n = 25
Formula: Sₙ = n/2 [2a + (n - 1)d]
Calculation:
S₂₅ = 25/2 [2(3) + (24)(4)]
S₂₅ = 25/2 [6 + 96]
S₂₅ = 25/2 × 102
S₂₅ = 25 × 51
Answer: S₂₅ = 1275
Formula: Sₙ = n/2 [2a + (n - 1)d]
Calculation:
S₂₅ = 25/2 [2(3) + (24)(4)]
S₂₅ = 25/2 [6 + 96]
S₂₅ = 25/2 × 102
S₂₅ = 25 × 51
Answer: S₂₅ = 1275
Example 3: Board-Level Question - 5 Marks
Question: The sum of 4th and 8th terms is 24, and sum of 6th and 10th terms is 44. Find first three terms.
Step 1: Express terms
a₄ = a + 3d, a₈ = a + 7d
a₆ = a + 5d, a₁₀ = a + 9d
Step 2: Set equations
Eq (i): 2a + 10d = 24 → a + 5d = 12
Eq (ii): 2a + 14d = 44 → a + 7d = 22
Step 3: Solve
Subtract: 2d = 10 → d = 5
From (i): a = 12 - 25 = -13
First three terms: -13, -8, -3
a₄ = a + 3d, a₈ = a + 7d
a₆ = a + 5d, a₁₀ = a + 9d
Step 2: Set equations
Eq (i): 2a + 10d = 24 → a + 5d = 12
Eq (ii): 2a + 14d = 44 → a + 7d = 22
Step 3: Solve
Subtract: 2d = 10 → d = 5
From (i): a = 12 - 25 = -13
First three terms: -13, -8, -3
6. Smart Tricks for CBSE Class 10 Board Exams
⚡ Trick 1: Quick nth Term Shortcut
If you know aₖ and need aₙ: aₙ = aₖ + (n - k)d
Example: If 10th term = 50, d=3, then 15th term = 50 + 5×3 = 65
⚡ Trick 2: Fast Sum for Odd n
When n is odd: Sₙ = n × (middle term)
Example: Sum of 2,5,8,11,14 = 5 × 8 = 40
⚡ Trick 3: AP Validity Check
Sequence is AP if: 2aₙ = aₙ₋₁ + aₙ₊₁
⚡ Trick 4: RAPID Method for Word Problems
- Read carefully
- Assign variables
- Pick formula
- Insert values
- Double-check
7. Visual Learning: Diagrams & Charts
Increasing vs Decreasing AP Comparison
| Feature | Increasing AP | Decreasing AP | Constant AP |
|---|---|---|---|
| Common Difference | d > 0 | d < 0 | d = 0 |
| Example | 3, 7, 11, 15... | 20, 15, 10, 5... | 5, 5, 5, 5... |
| Graph Trend | Upward slope | Downward slope | Horizontal line |
Flowchart for Solving AP Problems
START
↓
What is asked?
↓
Specific Term?
→ aₙ = a+(n-1)d
→ aₙ = a+(n-1)d
Sum of Terms?
→ Sₙ formula
→ Sₙ formula
Number of Terms?
→ n = [(l-a)/d]+1
→ n = [(l-a)/d]+1
8. Most Important Board Questions
⭐ 1 Mark Questions (MCQs)
Q1. Which term of AP: 21, 18, 15,... is zero?
Ans: 8th term (a₈ = 21 + 7(-3) = 0)
Ans: 8th term (a₈ = 21 + 7(-3) = 0)
Q2. Find common difference of AP: 1/2, 1/3, 1/6, 0...
Ans: d = -1/6
Ans: d = -1/6
⭐⭐ 2-3 Mark Questions
Q. Find sum of first 15 multiples of 8.
Solution: AP: 8, 16, 24...; a=8, d=8, n=15
S₁₅ = 15/2 [16 + 112] = 15/2 × 128 = 960
Solution: AP: 8, 16, 24...; a=8, d=8, n=15
S₁₅ = 15/2 [16 + 112] = 15/2 × 128 = 960
⭐⭐⭐⭐ 4-5 Mark Questions
Q. The sum of 5th and 9th terms is 72, and sum of 7th and 12th terms is 97. Find the AP.
Solution:
a₅ + a₉ = 72 → 2a + 12d = 72 → a + 6d = 36
a₇ + a₁₂ = 97 → 2a + 17d = 97
Solving: d = 5, a = 6
AP: 6, 11, 16, 21...
Solution:
a₅ + a₉ = 72 → 2a + 12d = 72 → a + 6d = 36
a₇ + a₁₂ = 97 → 2a + 17d = 97
Solving: d = 5, a = 6
AP: 6, 11, 16, 21...
9. Word Problems Section
Problem 1: Salary Increment
Question: A man starts with monthly salary ₹10,000 and annual increment ₹500. When will salary reach ₹15,000? What is total earnings then?
Solution:
AP: 10000, 10500, 11000... (a=10000, d=500)
Part 1: 15000 = 10000 + (n-1)500
5000 = (n-1)500 → n = 11 years
Part 2: S₁₁ = 11/2 [20000 + 5000] = 11/2 × 25000
Total earnings = ₹1,37,500
AP: 10000, 10500, 11000... (a=10000, d=500)
Part 1: 15000 = 10000 + (n-1)500
5000 = (n-1)500 → n = 11 years
Part 2: S₁₁ = 11/2 [20000 + 5000] = 11/2 × 25000
Total earnings = ₹1,37,500
Problem 2: Stadium Seating (CBSE Favorite)
Question: First row has 20 seats, second 24, third 28... Find seats in 15th row and total in 25 rows.
Solution: a=20, d=4
15th row: a₁₅ = 20 + 14×4 = 20 + 56 = 76 seats
Total 25 rows: S₂₅ = 25/2 [40 + 96] = 25 × 68 = 1700 seats
15th row: a₁₅ = 20 + 14×4 = 20 + 56 = 76 seats
Total 25 rows: S₂₅ = 25/2 [40 + 96] = 25 × 68 = 1700 seats
10. Common Mistakes to Avoid
Mistake 1: Forgetting (n-1)
Wrong: aₙ = a + nd
Correct: aₙ = a + (n-1)d
Mistake 2: Using Wrong Sum Formula
Don't use Sₙ = n/2(a+l) when last term is not given!
Mistake 3: Calculation Errors with Fractions
Always find common denominator: 1/3 - 1/2 = (2-3)/6 = -1/6
Mistake 4: Incomplete Steps (CBSE Penalty -1)
Always write: Formula → Substitution → Calculation → Final Answer
11. Practice Section
Multiple Choice Questions
Q1. If 7×7th term = 11×11th term, then 18th term is:
(a) 7 (b) 11 (c) 18 (d) 0 ✓
Q2. Sum of first n natural numbers is:
(a) n(n-1)/2 (b) n(n+1)/2 ✓ (c) n² (d) n(n+1)
(a) 7 (b) 11 (c) 18 (d) 0 ✓
Q2. Sum of first n natural numbers is:
(a) n(n-1)/2 (b) n(n+1)/2 ✓ (c) n² (d) n(n+1)
Case Study Question (4 Marks)
Context: TV production was 600 sets in 3rd year and 700 in 7th year. Production increases uniformly.
(a) Production in 1st year = 550 sets
(b) Production in 10th year = 775 sets
(c) Total production in 10 years = 6625 sets
(a) Production in 1st year = 550 sets
(b) Production in 10th year = 775 sets
(c) Total production in 10 years = 6625 sets
12. Frequently Asked Questions
Q1: Is Arithmetic Progressions important for CBSE Class 10?
Yes! This chapter carries 4-5 marks in CBSE Board Exams. It appears in both Standard and Basic Mathematics papers. Mastery of AP concepts is essential for scoring above 90%.
Q2: Which formula is most used in board exams?
The nth term formula (aₙ = a + (n-1)d) appears in 70% of questions. The sum formula is equally important for 3-5 mark questions. Memorize both!
Q3: Are these CBSE Class 10 Maths Notes based on NCERT?
Absolutely! These notes are 100% aligned with NCERT Class 10 Mathematics Chapter 5, covering all exercises 5.1 to 5.4 plus previous year board questions.
Q4: How to download Class 10 Maths Notes PDF?
These notes are optimized for both online reading and PDF conversion. You can save this page or use browser's print to PDF feature for offline study.
Best of Luck for Your CBSE Class 10 Board Exams 2025-26!
Master these CBSE Class 10 Notes and score 95%+ in Mathematics!
Next Chapter: Triangles – CBSE Class 10 Notes
Keywords: CBSE Class 10 Notes, CBSE Class 10 Maths Notes, Class 10 Maths Chapter 5 Notes, Arithmetic Progressions Class 10 Notes, AP Class 10 Notes, NCERT Class 10 Maths Notes, Class 10 Maths Notes PDF
