NCERT Solutions For Class 10 Maths Chapter 5 Arithmetic Progressions

What is Arithmetic Progression?

In simple terms, Arithmetic Progression or AP can be defined as a sequence of numbers. This sequence exists in an order in which the difference between any two consecutive numbers would be constant.

For example, if 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 is a series of natural numbers. It can be said that this series is also an arithmetic progression between the difference between every two successive terms is 1. If we have a similar series of odd and even numbers, still, the difference between two successive terms will be two. This means that the odd and even number series will also be arithmetic progressions.

It should also be noted for writing NCERT Solutions for Class 10 Maths Chapter 5 that we all observe examples of arithmetic progressions even in our daily lives. Some examples from the real-life of Arithmetic Progression Class 10 Maths Chapter 5 include roll numbers of students in a class, days of a week, and months in a year.

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

In Ch 5 Maths Class 10, it is also mentioned that there are three different types of progressions. These different types of progressions that are mentioned in Chapter 5 Class 10 Maths NCERT book are:

• Arithmetic Progression (AP)

• Geometric Progression (GP)

• Harmonic Progression (HP)

Before moving forward with the topic of Ch 5 Class 10 Maths, every student should know that a progression can be explained as a special type of sequence for which it is possible for one to obtain a formula for the nth term. When it comes to the subject of mathematics, then arithmetic progression is the most commonly used sequence.

There are also three other definitions of arithmetic progression that students should remember for writing Arithmetic Progression Class 10 NCERT Solutions. These definitions are:

• Definition One: Arithmetic progression is a mathematical sequence in which the difference between any two consecutive terms is always a constant. It can also be abbreviated as AP

• Definition Two: It is also mentioned in NCERT Solutions Class 10 Maths Chapter 5 that arithmetic progression or sequence is the sequence of numbers. In this sequence of consecutive numbers, it is possible to find the next number by adding a fixed number to the previous number in the chain

• Definition Three: In Arithmetic Progression Class 10 Solutions, the common difference of the AP is the fixed number that one should add to any term of the arithmetic progression. For example, in the arithmetic progression 1, 4, 7, 10, 13, 16, 19, 22, the value of the common difference is 3

It is important for students to also learn about the topic of notations in Class 10 Chapter 5 Maths. In Class 10 Maths Chapter 5 Solutions, there are three main terms. These terms are:

• The common difference (d)

• nth Term (an)

• The num of the first n terms (Sn)

These three terms are used in Class 10 Ch 5 Maths to represent the property of arithmetic progression. In the next section, we will look at these three properties in more detail.

• The Common Difference in Arithmetic Progression

According to ch 5 maths class 10 NCERT solutions, for any given series of arithmetic progression, the terms that are used are the first term, the common difference between any two terms, and the nth term. Let’s assume that a1, a2, a3, a4, …, an is an arithmetic progression.

This means that the value of the common difference ‘d’ is:

D = a2 - a1 = a3 - a2 = … = a1 - an-1

Here, d is the value of the common difference. The value of d can be positive, negative, or zero.

• The First Term of Arithmetic Progression

If an individual wants to write the arithmetic progression in terms of its common difference for solving an NCERT class 10 maths chapter 5 question, then it can be written as:

A, a + d, a + 2d, a + 3d, a + 4d, a + 5d, …, a + (n - 1) d

In this sequence, a is the first term of the progression.

The General Form of an Arithmetic Progression

In this section, students will be able to do just that. Before we proceed, a student should begin with an assumption that the arithmetic progression for class 10 maths ch 5 solutions is a1, a2, a3, …, an

The Formulas

Students often have to write Class 10 Maths Ncert Solutions Chapter 5 on the basis of the formulas that they learn from the chapter. Till now, we haven’t really looked at the formulas that are required for writing NCERT Solutions for Class 10 Maths Ch 5. That is about to change now as we will look at the various formulas that one will need for finding out Chapter 5 Maths Class 10 NCERT Solutions.

According to experts who write high-quality Class 10 Ch 5 Maths NCERT Solutions, there are mainly two formulas. If a student knows about both these formulas, then he or she will be able to write the majority of NCERT Solutions Of Class 10 Maths Chapter 5. These formulas are:

• The nth Term of Arithmetic Progression (AP)

This formula can be used for finding the class 10 maths chapter 5 NCERT solutions in which one needs to get the value of the nth term of an arithmetic progression. The formula can be written as:

An = a + (n - 1) d

Here, a is the first term, d is the value of the common difference, n is the number of terms, and an is the nth term.

Let’s take an example. Try to find out the nth term of the following arithmetic progression 1, 2, 3, 4, 5, …, an. The total number of terms is 15.

We know that n = 15. This means that according to the formula, we can say that:

An = a + (n - 1) d

Since, a = 1, and the common difference or d = 2 - 1 = 1

Then, an = 1 + (15 - 1) 1 = 1 + 14 = 15.

It should also be noted by students who refer to the NCERT Class 10 Maths Chapter 5 Solutions that the finite portion of an arithmetic progression is known as finite arithmetic progression. This means that the sum of a finite AP is known as an arithmetic series.

The behaviour of the entire sequence will also depend on the values of the common difference. This means that if the value of the common difference is positive, then the member terms will grow towards positive infinity. And if the value of the common difference is negative, then the member terms will move towards negative infinity.

• The Sum of the First n Terms

One can easily calculate the sum of n terms of any known progression. For an arithmetic progression, it is possible to calculate the sum of the first n terms if the value of the first term and the total terms are known. The formula is mentioned below.

S = n / 2 [2 a + (n - 1) x d]

But what if the value of the last term of the arithmetic progression is given? In that case, students should use the formula that is mentioned below.

S = n / 2 (first term + last term)

For ease of revision, we have also summarized all the major formulas of this chapter in a table. That table is mentioned below.

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