NCERT Solutions for Class 10 Maths Chapter 5: Arithmetic Progressions

Chapter 5 Arithmetic Progressions is one of the most important chapters in CBSE Class 10 Mathematics. It belongs to the Algebra unit, which carries 20 marks in total. AP alone typically contributes around 5 marks in the board exam, often appearing as one MCQ, one short-answer question, and sometimes as part of a case study. The chapter builds on the concept of sequences and patterns that students have encountered in earlier classes, formalising them into a powerful algebraic framework.

What Is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence 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 and is denoted by d. For example, the sequence 2, 5, 8, 11, 14, ... is an AP with first term a = 2 and common difference d = 3. APs appear everywhere in daily life - the fare structure of a taxi that charges a base fare plus a fixed amount per kilometre, the number of seats in successive rows of a stadium, monthly savings that increase by a fixed amount, and salary increments that follow a fixed annual raise. Not every sequence is an AP. The sequence 1, 4, 9, 16, ... (perfect squares) is not an AP because the differences between consecutive terms are 3, 5, 7, ... which are not constant. Similarly, a geometric sequence like 2, 6, 18, 54, ... has a constant ratio but not a constant difference.

Common Difference

The common difference d of an AP is found by subtracting any term from the term that follows it: d = a_(n+1) - a_n. This value remains the same throughout the AP. The common difference can be positive (increasing AP like 3, 7, 11, 15, ...), negative (decreasing AP like 20, 17, 14, 11, ...), or zero (constant AP like 5, 5, 5, 5, ...). To check whether a given sequence forms an AP, compute the differences between consecutive terms. If all differences are equal, it is an AP. For example, for the sequence -2, 2, 6, 10: differences are 4, 4, 4, so it is an AP with d = 4. For the sequence 1, 1, 2, 3, 5: differences are 0, 1, 1, 2, which are not constant, so it is not an AP.

The nth Term Formula

The nth term (or general term) of an AP is given by a_n = a + (n-1)d, where a is the first term, d is the common difference, and n is the position of the term. This formula allows you to find any term of the AP without listing all previous terms. For example, to find the 50th term of the AP 3, 7, 11, ...: a_50 = 3 + 49(4) = 3 + 196 = 199. The formula also helps in reverse. If you know a particular term, you can find the position. For instance, which term of the AP 5, 9, 13, ... is 101? Setting 101 = 5 + (n-1)(4) gives n = 25. Another common application is finding a term from the end of a finite AP. If the last term is l, the nth term from the end is l - (n-1)d. The nth term formula is also used to check whether a given number belongs to an AP. If solving for n gives a positive integer, the number is a term of the AP; otherwise, it is not.

Sum of First n Terms

The sum of the first n terms of an AP is denoted by S_n and is calculated using two equivalent formulas. When the first term a and common difference d are known: S_n = n/2[2a + (n-1)d]. When the last term l is known: S_n = n/2(a + l). The second formula is faster when the last term is directly available. For example, the sum of the first 20 terms of the AP 1, 4, 7, 10, ...: S_20 = 20/2[2(1) + 19(3)] = 10(2 + 57) = 10(59) = 590. A useful relationship connects S_n and a_n: a_n = S_n - S_(n-1) for n >= 2, and a_1 = S_1. This means if you are given S_n as a formula in terms of n, you can extract the nth term. For example, if S_n = 3n^2 + 5n, then a_n = S_n - S_(n-1) = 3n^2 + 5n - 3(n-1)^2 - 5(n-1) = 6n + 2. Problems involving S_n often lead to quadratic equations. Always solve for n using the quadratic formula or factoring and reject any negative or non-integer solution since n must be a positive integer.

Word Problems on AP

Word problems form a significant portion of CBSE board questions on AP. Common types include salary or savings problems where amounts increase by a fixed value each month or year, production problems where output increases uniformly, seating arrangements where each row has a fixed number of extra seats, penalty problems where fines increase daily by a fixed amount, and stacking problems where layers decrease uniformly. The key to solving word problems is identifying the first term a, the common difference d, and whether the question asks for a_n (a specific term) or S_n (total sum). For example, if a contractor faces a penalty of Rs 200 for the first day of delay and Rs 50 more for each subsequent day, the penalties form an AP with a = 200, d = 50. If the total penalty is Rs 27750, using the sum formula gives the number of days of delay.

How Tested in Board Exams

In CBSE board exams, AP questions typically appear in several formats. MCQs (1 mark each) test basic concepts like finding d, the nth term, or the sum. Short-answer questions (2-3 marks) involve finding specific terms, the number of terms, or straightforward sums. Long-answer questions (3-4 marks) feature word problems or multi-step calculations. Since 2024, case study questions (4-5 marks) on AP have appeared, usually set in real-life contexts like stadium seating, savings plans, or production scenarios. Assertion-reason questions test conceptual understanding of AP properties. The most frequently asked question types are finding the number of two-digit numbers divisible by a given number, finding the sum of first n terms, word problems on savings or production, and finding k when three expressions involving k are in AP.

Tips for Full Marks

To score full marks on AP questions, always write the formula before substituting values, as CBSE awards marks for showing the method. Verify your answer by back-substitution when possible. For sum problems that yield two values of n, check both and explain why one is rejected. In word problems, clearly define what a, d, and n represent. Use S_n = n/2(a + l) whenever the last term is known, as it reduces calculation. For MCQs, the shortcut d = coefficient of n in a linear a_n expression saves time. Always double-check the sign of d in problems involving decreasing sequences.

Connection to Other Chapters

Arithmetic Progressions connects to several other chapters in Class 10 Mathematics. The quadratic equations from Chapter 4 are needed when solving S_n problems that lead to quadratic equations. Coordinate geometry (Chapter 7) uses AP when dealing with equally spaced collinear points. In statistics (Chapter 13), the assumed mean method for finding the mean of grouped data uses concepts related to equally spaced class marks that form an AP. Understanding AP also lays the foundation for geometric progressions and series studied in Class 11. The algebraic skills of forming and solving equations, which run through Chapters 3, 4, and 5, are all interconnected.