NCERT Solutions For Class 10 Maths Chapter 5 Arithmetic Progressions

You can opt for Chapter 5 - Arithmetic Progressions NCERT Solutions for Class 10 Maths PDF for Upcoming Exams  and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 10 Maths

• Chapter 1 - Real Numbers

• Chapter 2 - Polynomials

• Chapter 3 - Pair of Linear Equations in Two Variables

• Chapter 4 - Quadratic Equations

• Chapter 5 - Arithmetic Progressions

• Chapter 6 - Triangles

• Chapter 7 - Coordinate Geometry

• Chapter 8 - Introduction to Trigonometry

• Chapter 9 - Some Applications of Trigonometry

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Areas Related to Circles

• Chapter 13 - Surface Areas and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Arithmetic Progression

Have you ever wondered why are numbers so wondrous? Ch 5 Arithmetic Progressions in NCERT Class 10 gives you the brief idea of one of the most wonderful number patterns in Mathematics. It takes you through the definition, examples and few other concepts related to the Arithmetic progressions. This chapter is divided into four sections and three exercises. 

List of Exercises and Topics they Cover:

• Exercise 5.1 : Arithmetic Progression.

• Exercise 5.2 : nth Term Of An AP.

• Exercise 5.3 : Irrational Numbers

• Exercise 5.4 : Rational Numbers and their decimal expansions.

The Fantasy of Numbers: Arithmetic Progression:

We come across an infinite number of patterns of Mathematics in our day to day life. Most of us just cherish at the sight of these patterns and move one while only a few wonder “WHAT IS THE MATHEMATICS BEHIND THIS PATTERN!!” One such amazing number pattern is described in Chapter 5 Maths Class 10. Arithmetic progressions is one of the most important and interesting topics in the Class 10 NCERT text book. The chapter describes the definition of arithmetic progression with examples. It also explains the general form of representing arithmetic progressions. The chapter also covers an ample number of problems which explains the steps to determine the nth term of an arithmetic sequence and the sum of the terms in an arithmetic sequence.

Exercise 5.1: Introduction and Arithmetic Progression

Exercise 5.2: Nth term of an Arithmetic Progression

Exercise 5.3: Sum of the terms in Arithmetic Progression

Exercise 5.4: Additional questions

Sum and Substance of Ch 5 Class 10 Maths:

• Arithmetic progression is a sequence of numbers in which the difference between any two consecutive numbers is a constant when the numbers are taken in the same order. As the difference between the terms is constant throughout the sequence, it is called the common difference. 

• An Arithmetic progression is written in a general form as:

a, a + d, a + 2d, a + 3d, …….., a + (n-1)d

In this general form, ‘a’ is the first term in the sequence, ‘d’ is the common difference and ‘n’ is the number of terms in the sequence. The first term and common difference may take up negative values. However, the number of terms in the sequence can never be a negative integer. The number of terms is always a positive whole number.

• When the first term ‘a’ and the common difference ‘d’ is known in an arithmetic sequence with ‘n’ terms, the nth term of this sequence is calculated as:

an = a + (n - 1) d

• The sum of the terms in an arithmetic sequence of ‘n’ terms with ‘a’ as the first term, ‘d’ as the common difference is computed as:

Sₙ = \[\frac{n}{2}\] [2a + (n-1)d]

• The Arithmetic progression with ‘a’ as the first term and ‘l’ as the last term, the sum of the terms in the AP is given by product of half the number of terms in the AP and the sum of first and last terms.

Sₙ = \[\frac{n}{2}\] [a + l]

• The arithmetic mean of any set of numbers is the average of the numbers.

Interesting Fact About Arithmetic Progression Class 10:

Johann Carl Friedrich Gauss introduced the concept of Arithmetic progression. The victorious story of invention goes like this. During his schooling, the Mathematics teacher asked all the students to calculate the sum of the first 100 positive integers. He assigned this work to the students so that he can keep them engaged with work and take some rest. However, Gauss came up with the perfect answer to the question in a few moments. When asked by the teacher to explain the procedure he followed to arrive at the answer, Gauss explained it. He has arranged the first 100 integers taken in ascending order as one row and in descending order as the other row. When each corresponding element of each row was added, the sum was found to be 101. As the number of elements was hundred, he calculated the total sum as 100 x 101 = 10100. However, since he had taken the set of 100 integers twice, he divided the sum obtained by 2 and arrived at the final answer of the sum of 100 integers as 10100/2 = 5050. This formed the basis for all the further advances in the concept of arithmetic progression. 

NCERT Solutions for Class 10 Chapter 5 PDF by Vedantu Stands out to be Unique:

• The NCERT Solutions for Class 10 Chapter 5 is devised with utmost care and we at Vedantu ensure that you get the best input while referring to the solutions.

• Choice of words and vocabulary is also given due importance so that no incorrect problem solving methodology is conveyed due to grammatical errors. 

• Students and teachers are benefited with step by step solutions to the problem so that you can understand the solution easily. 

• The highly experienced academicians at Vedantu have always been successful in recreating unique study material and NCERT Solutions for Class 10 Maths Chapter 5 pdf and also outstanding solutions for other chapters.

• We always put in our best effort to foster top rated accurate solutions to all the questions considering the CBSE pattern and marking scheme for each answer.

Why Learn About Arithmetic Progression Class 10!!!

Arithmetic progression is one of the most important chapters in NCERT Class 10 Mathematics. It gives a brief overview of one of the most important number patterns used in day to day life. Arithmetic progressions are also of due importance in real life. The best example of arithmetic progression is the computation of bank balance when simple interest is added on a monthly basis. With respect to CBSE Class X Board Examination,Chapter 5 Maths Class 10 is very important for scoring the best marks. The chapter contributes for the 5% of the total marks in the Board Examination. Generally, 2 objective type or very short answer questions and one short answer question type II is reflected in the question paper. With constant revision and effective practice, any questions from this chapter can be easily cracked to gain full marks. NCERT Solutions for Class 10 Maths Chapter 5 will assist you to understand the solutions to all the exercise problems step by step and will in turn make you more confident to answer any questions from this chapter with all twists and turns.

What does the Chapter 5 Arithmetic Progressions explain?

Fantasy of Numbers: Numbers Patterns

Numbers are the basic building blocks of Mathematics. These numbers are fantastic and can give wondrous patterns when arranged with a little care. Hundreds of funny number patterns can be created with a little knowledge of the basics of Mathematics. These patterns in Mathematics enhance the logical thinking ability of the students and other learners. These patterns also help the biologists to predict the behavior of several living organisms. Meteorologists can foresee tornadoes, thunderstorms and hurricanes. The number patterns find an impeccable importance in all areas of scientific research.

Arithmetic Progression: An Amazing Pattern

Arithmetic progression is a number pattern in Mathematics in which there exists a common difference between the consecutive elements of the sequence. You can compute the next element in the sequence by adding the value of common difference to the preceding element in the sequence. Each element in the sequence can be represented in terms of the first term of the sequence and the common difference.

Second Term = First term + Common Difference

Third term = Second term + Common Difference 

= First term + Common difference + Common difference

Third term = First term + 2 (Common Difference)

If the first term in the sequence is denoted as ‘a’ and the common difference in the AP is ‘d’, then the AP is represented as:

AP = a, (a + d), (a + d + d), (a + d + d + d), ……

AP = a, (a + d), (a + 2d), (a + 3d), …….

The difference between two consecutive terms remains the same throughout the sequence. Hence it is called the common difference. In the sequence, 

• The difference between second and first term is a + d - a = d

• The difference between third and second term is a + 2d - (a + d) = d and so on.

So, the difference between any pair of immediate terms remains the same in the arithmetic progression.

Arithmetic progression Class 10 with countable number of elements is called the finite arithmetic progression and the arithmetic progression with infinite number of elements is called the infinite arithmetic progression. 

Wondering How to Find Terms in An AP!!!

Any term in an arithmetic progression can be computed when the first term of the sequence and the common difference are known. The nth term of the sequence is obtained by adding the first term to the product obtained when one less than the term number is multiplied with the common difference. 

• First term of the arithmetic sequence is ‘a’ and can be rewritten as: a + (0 x d).

• Second term of the arithmetic sequence can be rewritten as: a + (1 x d).

• Third term of the arithmetic sequence can be rewritten as: a + (2 x d).

• nth term of the arithmetic sequence can be rewritten as: a + (n - 1) d.

So, the nth term of an arithmetic sequence can be found using the formula, 

aₙ = a + (n-1)d

In the above equation, an is the nth term, ‘n’ responds to the position of the term an, ‘a’ is the first term and ‘d’ is the common difference. 

Terms as a Whole: Sum of the Terms in AP

It is important to know how to find the sum of the given number of terms in an arithmetic progression. Finding the sum of ‘n’ terms in an AP is used in several Mathematical applications. The sum of ‘n’ terms in an AP is calculated as the product obtained when half the number of terms in the AP is multiplied with the product of one less than the number of terms multiplied by the common difference subtracted from twice the first term in the sequence. The formula to find the sum of ‘n’ terms in an arithmetic sequence is:

Sₙ = \[\frac{n}{2}\] [2a + (n-1)d]

In the above equation, Sₙ is the sum of ‘n’ terms in the sequence, ‘n’ is the number of terms, ‘a’ is the first term and ‘d’ is the common difference.

The above formula can be rewritten as:

Sₙ = \[\frac{n}{2}\] [a + a + (n-1)d]

But, the nth term in an AP is given as 

aₙ = a + (n-1)d

Sₙ = \[\frac{n}{2}\] [a + aₙ]

If an is the last term in the sequence denoted as ‘l’, then the sum of the terms is given as the product of half the number of terms and the sum of the first and last terms.

Sₙ = \[\frac{n}{2}\] [a + l]

The Arithmetics of Arithmetic Progression Class 10:

Ch 5 Class 10 Maths consist of crisp and clear information and problems related to Arithmetic Progression. We, at Vedantu, make sure that the CBSE textbook content is paraphrased considering the age group of the students and also help the teaching faculties to convene the students with appropriate data. This chapter consists of 4 different sections which conveniently elaborates the concepts related to arithmetic progression along with an ample of examples and exercise questions. The brief summary of the exercises in the chapter is given below.

Exercise 5.1: Introduction and Arithmetic Progression

The exercise consists of 4 questions with many sub questions related to the identification and general form of representation of arithmetic sequences. The problems also give a clear idea of identifying whether a given sequence is an arithmetic progression or not.

Exercise 5.2: Nth term of an Arithmetic Progression

The second exercise comprises 20 questions. Each question is related to finding the nth term or the number of terms in a sequence. The questions cover all domains of bloom’s taxonomy including understanding, application and creation. 

Exercise 5.3: Sum of the terms in Arithmetic Progression

This exercise consists of 20 questions which involves the determination of the sum of all the terms in a sequence when the sequence is in arithmetic progression. 

Exercise 5.4: Additional Questions

The exercise has a set of 5 questions that are of higher order thinking skills. These questions are optional and are left to the interest of students. Moreover, these questions will not appear for the CBSE board examination. 

Deeper into the Exercises:

Section 5.1 and 5.2: Exercise 5.1

In the sections 5.1 and 5.2 of NCERT Class 10 Maths Chapter 5, the basic introduction and the general representation of an arithmetic sequence is briefly discussed and the examples for arithmetic progression are also stated to make the concept more clear and feasible to you. 

The exercise corresponding to these sections consists of problems related to identification of arithmetic patterns from the given set of patterns, finding the first few terms of the arithmetic sequence for which the first term and the common difference are given and determining the first term & common difference from the given arithmetic sequence. 

Section 5.3: Exercise 5.2

The exercise corresponding to this section in NCERT Class 10 Chapter 5 has 10 example problems and 20 exercise questions for practice. There are a number of practice questions to find the first term / common difference / nth term / number of terms (n) in the arithmetic sequence when any three of the four values are given. It also includes many real life world problems that involve finding the term at a given position in the arithmetic sequence. For example, suppose the annual increment of a person’s salary is 5% of the initial salary, then the salary of the person after a few years can be calculated using the formula for nth term in a sequence. 

Section 5.4: Exercise 5.3

This section in NCERT Class 10 Chapter 5 consists of 5 example problems and 20 practice questions. Each word problem related to finding the sum of terms in an AP is unique and based on different worldly situations. It also involves the problems which turn out to be mathematical formulas such as finding the sum of first ‘n’ natural numbers, first ‘n’ odd or even numbers etc. 

Sum of the first ‘n’ natural numbers is n (n + 1) / 2. The exercise also involves a number of problems in which a term is given. We can determine whether the given term is a part of the given arithmetic sequence or not. 

Exercise 5.4

There is an additional exercise consisting of 5 exemplar problems. These problems pertain to higher order thinking skills. It triggers the understanding of the students about the concept. Though this exercise is not compulsory to be solved by the teacher in class, we always recommend the students and teachers to know the problem solving methodology for these questions. NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions give the step by step solutions to these optional problems also. You can enhance your conceptual understanding about the Arithmetic progression Class 10 by solving the problems in this additional exercise. 

Arithmetic Progression Class 10 in a Nutshell:

• Arithmetic progression is a sequence of numbers in which the consecutive terms in the sequence can be determined by adding the same integer to the sequence. This common integer is called the common difference.

• The first element of the arithmetic sequence is generally represented by the letter ‘a’ and the common difference is represented by the letter ‘d’. ‘n’ is the number of terms in the sequence and ‘an and Sn’ are the value of the nth term and the value of sum of the terms in the arithmetic sequence respectively.

• The formula to find nth term of the sequence is aₙ = a + (n-1)d.

• The formula to find the sum of the terms in an AP is Sₙ = \[\frac{n}{2}\] [2a + (n-1)d].

• These formulas can be used to determine whether the given number is a term of the given AP or not when the nth terms and sum of the terms are given.

• This chapter is of utmost importance because it covers 5% of the total marks in the Class 10 Mathematics examination.