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Class X Mathematics
Chapter 5: Arithmetic Progressions
Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.
Syllabus Sections
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Chapter Overview
Welcome to Class X Mathematics: Arithmetic Progressions. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.
Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.
Prerequisite Concepts
Play with PatternsSimple Equations
About This Chapter
This comprehensive study guide for Arithmetic Progressions is designed for Class X students following the CBSE and NCERT Mathematics curriculum. It covers 5 key subtopics including Arithmetic Progression definition, Common difference d, n-th term of an AP formula, and 2 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.
The chapter includes 2 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.
Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Arithmetic Progressions, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.
What You'll Learn in This Chapter
By the end of studying Arithmetic Progressions for Class X, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:
Identify if a sequence is an Arithmetic Progression.
Calculate any term (a_n) using first term and common difference.
Compute total sum of sequences to solve real-world salary or saving steps.
Prerequisites for This Chapter
Before studying Arithmetic Progressions, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:
Play with PatternsSimple Equations
If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Arithmetic Progressions with full confidence.
Real-World Applications of Arithmetic Progressions
Students often wonder “Where will I use Arithmetic Progressionsin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:
Savings & Investment Plans
Fixed monthly deposits in a recurring deposit account form an arithmetic progression, making it easy to calculate total savings over time.
Stacking & Arrangement Patterns
Objects stacked in rows (like seats in a stadium) where each row has a fixed number more than the previous follow an AP pattern.
Salary Increments
Annual fixed-amount raises create an arithmetic sequence of yearly salaries, useful for career financial planning.
Construction Planning
Brick-laying patterns in pyramidal structures where each layer reduces by a constant number follow arithmetic progressions.
Understanding the real-world relevance of Arithmetic Progressions not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.
Study Tips for Arithmetic Progressions
Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Arithmetic Progressions:
✏️
Practice Step-by-Step
Write out every intermediate step when solving problems. Skipping steps is the most common source of errors in calculation-heavy chapters. Build speed only after achieving consistent accuracy.
🔄
Verify by Back-Substitution
After finding your answer, substitute it back into the original equation to verify correctness. This simple habit catches most arithmetic and sign errors before they cost you marks.
📝
Maintain an Error Log
Keep a dedicated notebook of mistakes you make during practice. Review it weekly to identify patterns — you will notice the same types of errors recurring and can actively work to eliminate them.
Pro Tip: Consistency beats intensity. Studying Arithmetic Progressions for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.
Detailed Topic Breakdown
Detailed Subtopics Study Guide
Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:
1Arithmetic Progression definition
Concept Explanation
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. Each term (except the first) is obtained by adding a fixed number to the preceding term.
Mathematical Representation
a, \, a+d, \, a+2d, \, a+3d, \, ...
Study Guideline: Verify if a sequence is an AP by checking if the difference between any term and its predecessor (a_n - a_{n-1}) is constant.
2Common difference d
Concept Explanation
The fixed number that is added to each term to get the next term in an AP is called the common difference. It can be positive, negative, or zero.
Mathematical Representation
d = a_2 - a_1 = a_3 - a_2 = a_n - a_{n-1}
Study Guideline: Calculate d by subtracting the first term from the second term (d = a2 - a1). Do not subtract in the wrong order.
3n-th term of an AP formula
Concept Explanation
The n-th term of an AP with first term 'a' and common difference 'd' is calculated using a linear formula, which allows us to find any term without writing the entire sequence.
Mathematical Representation
a_n = a + (n-1)d
Study Guideline: Identify the values of a, d, and n from the problem, substitute them into the formula, and solve. Note that 'n' must always be a positive integer.
4Sum of first n terms of an AP
Concept Explanation
The sum of the first n terms of an AP is the total accumulated sum of the terms. It can be calculated using the first term, common difference, and number of terms, or using the first and last terms.
Mathematical Representation
S_n = \frac{n}{2}[2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2}[a + l] \quad \text{where } l = a_n
Study Guideline: Use the formula involving 'l' (last term) if the last term is known, as it simplifies the calculation significantly.
5Applications word problems
Concept Explanation
Real-world problems applying area, perimeter, and circumference formulas to paths, fences, and circle tracks.
Mathematical Representation
\text{Area of Path} = \text{Area of Outer Shape} - \text{Area of Inner Shape}
Study Guideline: Draw a diagram to visualize outer and inner boundaries (e.g., a path around a rectangular field).
Learning Outcomes
By the end of this chapter, students will be able to master and solve questions on these outcomes:
Identify if a sequence is an Arithmetic Progression.
Calculate any term (a_n) using first term and common difference.
Compute total sum of sequences to solve real-world salary or saving steps.
Sub-Topics Directory
Arithmetic Progression definitionCommon difference dn-th term of an AP formulaSum of first n terms of an APApplications word problems
Key Formulas & Equations
Click Copy to duplicaten-th term of AP
a_n = a + (n-1)d
Finds the value of the n-th term.
Sum of n terms AP
S_n = \frac{n}{2}[2a + (n-1)d]
Sum of first n terms.
Related Formulas from Adjacent Chapters
The Quadratic Formula
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
[From Quadratic Equations] Finds the exact roots of ax² + bx + c = 0.
Discriminant
D = b^2 - 4ac
[From Quadratic Equations] Determines nature of roots (D > 0: two real roots; D = 0: one real root; D < 0: two complex roots).
Basic Proportionality Theorem
\frac{AD}{DB} = \frac{AE}{EC}
[From Triangles] If a line is parallel to one side of a triangle intersecting other sides, it divides them proportionally.
Interactive Problem Solver
**AP Term Calculator**
Find the 10th term of the AP: 2, 7, 12, ...
Scientific Calculator Active: A custom solver for the specific subtopics of this chapter is currently in development. In the meantime, please use our comprehensive Scientific Functions Calculator below to verify your study equations.
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Adjust Inputs
45
Calculated Results
Function Output
0.71
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Deterministic Mathematical Simulation Engine • Verified Calculations
Date: 9/1/2026
Time: 9:27:41 AM
Scientific Functions Calculator
Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.
1. Input Parameters
| Parameter | Value | Unit |
|---|---|---|
| Value (X) | 45 | — |
| Function | sin | — |
2. Calculated Results
| Metric | Calculated Output |
|---|---|
| Function Output | 0.707 |
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Student Solver & Visualizer Guide
Real-time calculationsStep-by-step solving
1. Collect input parameters:
- operand1 = 45
- operation = sin
2. Feed parameters through standard algorithmic calculation methods.
3. Compile calculated results:
- calcResult = 0.7071
Student-friendly explanations
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Class X Quiz Challenge
Test your math skills with this 5-question chapter mock exam. A live ticking clock will monitor your performance.
Class X CBSE Board Practice Questions
Review and solve 31 real Board paper questions from the 20 mock sample sets matching Arithmetic Progressions.
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Sample Paper 02 | Question 29Section C (3 Marks)
Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 7? (ii) 14? (iii) equal to 12?
Sample Paper 03 | Question 28Section C (3 Marks)
Two dice are thrown at the same t ime. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) 7? (iii) less than or equal to 12?
Sample Paper 04 | Question 3Section A (1 Mark)
For the following distribution: Class 0-5 5-10 10-15 15-20 20-25 Frequency 10 15 12 20 9 the sum of lower limits of the median class and modal class is
Sample Paper 05 | Question 5Section A (1 Mark)
The sum of the lower limit of median class and the upper limit of the modal class of the following data is: Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 No. of students 8 10 12 22 30 18
Sample Paper 05 | Question 18Section A (1 Mark)
Two different dice are thrown together. The probability of getting the sum of the two numbers less than 7 is:
Sample Paper 05 | Question 32Section D (5 Marks)
If the median of the following distribution is 58 and sum of all the frequencies is 140. What is the value of x and y? Class 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95 Frequency 8 10 x 25 40 y 15 7
Sample Paper 06 | Question 31Section C (3 Marks)
Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 5? (ii) 10? (iii) at least 9?
Sample Paper 07 | Question 28Section C (3 Marks)
A father’s age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
Sample Paper 07 | Question 38Section E (4 Marks)
Aditya is a fitness freak and great athlete. He always wants to make his nation proud by winning medals and prizes in the athletic activities. An upcoming activity for athletes was going to be organised by Railways. Aditya wants to participate in 200 m race. He can currently run that distance in 51 seconds. But he wants to increas e his speed, so to do it in 31 seconds. With each day of practice, it takes him 2 seconds less. (i) He wants to makes his best time as 31 sec. In how many days will be able to achieve his target? (ii) What will be the difference between the time taken on 5th day and 7th day. OR (ii) Which term of the arithmetic progression 3, 15, 27, 39 .... will be 120 more than its 21st term?
Sample Paper 09 | Question 31Section C (3 Marks)
(a) If the sum of first m terms of an A.P. is same as sum of its first n terms (m ≠ n), then show that the sum of its first (m + n) terms is zero.
Sample Paper 10 | Question 5Section A (1 Mark)
For the following distribution: Class 0-5 5-10 10-15 15-20 20-25 Frequency 10 15 12 20 9 the sum of lower limits of the median class and modal class is
Sample Paper 10 | Question 12Section A (1 Mark)
In an A.P., if the first term a = 7, nth term an 84 and the sum of first n terms S n = 2093/2, then n is equal to:
Sample Paper 10 | Question 15Section A (1 Mark)
The sum of the length, breadth and height of a cuboid is 6√3cm and the length of its diagonal is 2√3cm. The total surface area of the cuboid is
Sample Paper 10 | Question 32Section D (5 Marks)
The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.
Sample Paper 11 | Question 4Section A (1 Mark)
If the sum of the first n terms of an A.P. be 3n² + n and its common difference is 6, then its first term is
Sample Paper 11 | Question 19Section A (1 Mark)
Assertion (A): a, b, c are in A.P. if only if 2b = a + c. Reason (R): The sum of first n odd natural numbers is n2.
Sample Paper 11 | Question 27Section C (3 Marks)
The sum of two numbers is 15. If the sum of their reciprocals is 3/10, find the two numbers.
Sample Paper 12 | Question 10Section A (1 Mark)
The 11th and 13th terms of an AP are 35 and 41 respectively, its common difference is
Sample Paper 12 | Question 38Section E (4 Marks)
Mohan takes a loan from a bank for his car. Mohan replays his total loan of Rs.118000 by paying every month starting with the first instalment of Rs.1000. If he increases the instalment by Rs.100 every month. (i) What is the first term and common difference of given question. (1) (ii) The amount paid buy him in 30th instalment. (1) (iii) The amount paid by him in the 30 instalments is (2) (OR) (iii) What amount does he still have to pay after 30th instalment? (2)
Sample Paper 13 | Question 17Section A (1 Mark)
Two different dice are thrown together. The probability of getting the sum of the two numbers less than 7 is:
Sample Paper 14 | Question 6Section A (1 Mark)
If the sum of the circumferences of two circles with radii R1 and R2 is equal to the circumference of a circle of radius R, then:
Sample Paper 14 | Question 30Section C (3 Marks)
(i) Favourable outcomes for a card numbered not divisible by 3 = 20 (i.e. 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28 and 29). Probability of drawing a card numbered not divisible by 3 = 20/30 = 2/3 (ii) Favourable outcomes for a prime numbered card greater than 7 = 6 (i.e. 11, 13, 17, 19, 23 and 29) Probability of drawing a prime number card, greater than 7 = 6/30 = 1/5 (iii) Favourable outcomes for not a perfect square numbered card = 25 (leaving 1, 4, 9, 16 and 25) Probability of drawing a card which is not a perfect square = 25/30 = 5/6 27. Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that ∠APB = 2∠OAB. PA & PB are the tangents to a circle, with Centre O from a point P outside it. PA= PB. (The tangents to a circle from an external point are equal in length ) ∠PBA = ∠PAB [Angles opposite to the equal sides of a triangle are equal.] ∠APB+ ∠PBA +∠PAB= 180° [Sum of the angles of a triangle is 180°] x° + ∠PAB +∠PAB = 180° (∠PBA = ∠PAB) ⇒ x° + 2∠PAB = 180° ⇒ ∠PAB =½(180° - x°) ⇒ ∠PAB =90° - x°/2 ⇒ ∠OAB +∠PAB=90° ⇒ ∠OAB =90° - (90° - x°/2) ⇒ ∠OAB = x°/2 ⇒ ∠OAB = ∠APB /2 ⇒ ∠APB = 2∠OAB OR Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Sample Paper 15 | Question 13Section A (1 Mark)
Find the sum of all two-digit odd positive numbers.
Sample Paper 15 | Question 21Section B (2 Marks)
Find the sum of the last 10 terms of the AP: 8, 10, 12, ..., 126.
Sample Paper 15 | Question 36Section E (4 Marks)
Case Study 1: Ananya saves money uniformly for her daughter's higher education. She saves Rs 24 in the first week, Rs 30 in the second week, Rs 36 in the third week, and so on. (i) Verify if the savings form an AP. (ii) Find her savings in the 15th week. (iii) In which week will her savings be Rs 66?
Sample Paper 16 | Question 18Section A (1 Mark)
Find the sum of all natural numbers from 1 to 10.
Sample Paper 16 | Question 29Section C (3 Marks)
If the sum of first n terms of an AP is 5n^2 - 3n, find the 10th term.
Sample Paper 17 | Question 5Section A (1 Mark)
The common difference of the A.P. 1 1 4 1 8, , ,........2 2 2 x x x x x - - is
Sample Paper 18 | Question 4Section A (1 Mark)
The common difference of an A.P., if a23 - a19 = 32, is :
Sample Paper 19 | Question 24Section B (2 Marks)
The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 2 days
Sample Paper 19 | Question 32Section D (5 Marks)
The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Compute the missing frequencies f1 and f2. Class 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 5 f1 10 f2 7 8
Critical Mistakes to Avoid
Recognizing and eliminating common errors is a key step to scoring high marks in school and board exams:
!Order of Operations (BODMAS)
Solving additions or subtractions before evaluating parentheses, exponents, divisions or multiplications.
!Improper Algebraic Isolation
Doing addition/subtraction to only one side of the equation when rearranging terms.
!Decimal Place Errors
Misplacing the decimal dot when dividing by factors of ten.
Revision Summary Cards
1Formula Memorization
Commit core formulas to memory and write them down before solving problems.
2Units Consistency
Verify that all physical parameters use compatible measurement units.
3Calculator Check
Use our interactive calculator solvers to verify manually computed decimal answers.
4Prerequisites Checked
Ensure all prerequisite concepts are fully understood before working on advanced calculations.
5Step-by-Step Checking
Always check equations step-by-step to avoid simple sign and arithmetic errors.
Frequently Asked Questions
Resolve equations using alternative methods (e.g., check quadratic roots by plugging them back into ax² + bx + c), verify decimals using scientific solvers, and review step-by-step worked calculations.
Trigonometry is used in architecture, mechanics and navigation. Quadratic equations model rocket trajectories and profit boundaries. Basic arithmetic and fractions are applied in accounting, budgeting and lab chemistry.
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Print-Ready Math Worksheet
Generate and print a beautiful chapter worksheet complete with key equations, problems, and dedicated writing spaces.
Worksheet Details:
- • Title: Class X Math: Arithmetic Progressions Printable Sheet
- • Format: Print Optimized CSS stylesheet
- • Sections: Formulas checklist, 5 assessment questions, worked keys
NexPro Education Worksheet
Class X Mathematics: Arithmetic Progressions
Name: ______________________Date: __________________Score: ______ / 5
Section I: Key Formulas
n-th term of AP
a_n = a + (n-1)d
Sum of n terms AP
S_n = \frac{n}{2}[2a + (n-1)d]
Section II: Answer the Questions
1. Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 7? (ii) 14? (iii) equal to 12?
Solve and write the step-by-step solution in the workspace below.
2. Two dice are thrown at the same t ime. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) 7? (iii) less than or equal to 12?
Solve and write the step-by-step solution in the workspace below.
3. If the median of the following distribution is 58 and sum of all the frequencies is 140. What is the value of x and y? Class 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95 Frequency 8 10 x 25 40 y 15 7
Solve and write the step-by-step solution in the workspace below.
4. Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 5? (ii) 10? (iii) at least 9?
Solve and write the step-by-step solution in the workspace below.
5. A father’s age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
Solve and write the step-by-step solution in the workspace below.