Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

RD Sharma Class 10 Solutions Chapter 9 - Exercise 9.3

ffImage
Last updated date: 24th May 2024
Total views: 577.2k
Views today: 5.77k

RD Sharma Class 10 Solutions Chapter 9 - Arithmetic Progressions (Ex 9.3) Exercise 9.3 - Free PDF

Free PDF download of RD Sharma Class 10 Solutions Chapter 9 - Arithmetic Progressions Exercise 9.3 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 9 - Arithmetic Progressions Ex-9.3 Questions with Solutions for RD Sharma are available to help you to revise complete Syllabus and score more marks. Register for online coaching for IIT JEE (Mains & Advanced) and other Engineering entrance exams. Register Online for Class 10 Science tuition on Vedantu.com to score more marks in CBSE board examination.


Every NCERT Solution is provided to make the study simple and interesting on Vedantu. Vedantu.com is No.1 Online Tutoring Company in India, it provides you Free PDF download of NCERT Solutions for Class 10 Maths solved by Expert Teachers as per NCERT (CBSE) Book guidelines. All Chapter wise Questions with Solutions are available to help you to revise complete Syllabus and Score More marks in your examinations.

Popular Vedantu Learning Centres Near You
centre-image
Mithanpura, Muzaffarpur
location-imgVedantu Learning Centre, 2nd Floor, Ugra Tara Complex, Club Rd, opposite Grand Mall, Mahammadpur Kazi, Mithanpura, Muzaffarpur, Bihar 842002
Visit Centre
centre-image
Anna Nagar, Chennai
location-imgVedantu Learning Centre, Plot No. Y - 217, Plot No 4617, 2nd Ave, Y Block, Anna Nagar, Chennai, Tamil Nadu 600040
Visit Centre
centre-image
Velachery, Chennai
location-imgVedantu Learning Centre, 3rd Floor, ASV Crown Plaza, No.391, Velachery - Tambaram Main Rd, Velachery, Chennai, Tamil Nadu 600042
Visit Centre
centre-image
Tambaram, Chennai
location-imgShree Gugans School CBSE, 54/5, School road, Selaiyur, Tambaram, Chennai, Tamil Nadu 600073
Visit Centre
centre-image
Avadi, Chennai
location-imgVedantu Learning Centre, Ayyappa Enterprises - No: 308 / A CTH Road Avadi, Chennai - 600054
Visit Centre
centre-image
Deeksha Vidyanagar, Bangalore
location-imgSri Venkateshwara Pre-University College, NH 7, Vidyanagar, Bengaluru International Airport Road, Bengaluru, Karnataka 562157
Visit Centre
View More
Competitive Exams after 12th Science

General Vocabulary used in Arithmetic Progressions

While studying the arithmetic progression, we may come across the following terms. 


First Term: As given by its name, the first term in an AP is the first term in the progression. It is usually represented by a1 or simply a. For example, in sequence 2,4,6,8,10…. 2 is the first term. 


Common Difference: As mentioned above, an AP is a sequence where every term, except the first term, is obtained by adding a fixed number to its previous term. This fixed number is called the “common difference” and it is generally denoted by 'd'. If the first term is a1, then: the second term is a1+d, the third term is a1+d+d = a1+2d, and the fourth term is a1+2d+d= a1+3d and the sequence goes on. For example, in the sequence 2,4,6,8,10. . . Each term, except the first term which is 2, is obtained by adding 2 to its previous term. So, in this case, the common difference is, d=2. Generally, the common difference is the difference between every two successive terms of an Arithmetic progression. Hence, the formula to calculate the common difference of an AP is:  d=an-a{n-1}


The General Term of an Arithmetic Progression (nth Term)

The general term also called the nth term of an AP, whose first term is a and the common difference is d, is found by the formula an=a+(n-1)d. 

 

The Formula for Calculating Sum of Arithmetic Progression

Consider an arithmetic progression sequence whose first term is a1 or a and the common difference is d.

  • The sum of the first n terms of an arithmetic progression when the nth term is not known to us is Sn = n/2[2a+(n-1) d] 

 

  • The sum of the first n terms of an arithmetic progression when the nth term, an is known to us is Sn = n/2[a1+an


Derivation of the Arithmetic Progression Formula

To find the nth term of an arithmetic progression, we know that the nth term, an=a1+(n–1)d, where, a1 is the first term, a1 + d is the second term, the third term is a1 + 2d, and the sequence goes on. For finding the sum of the arithmetic series, say Sn, we start with the first term and successively add the common difference to the next coming terms.

 

Sn = a1 + (a1 + d) + (a1 + 2d) + … + [a1 + (n–1)d].

 

Also, we can start with the nth term and successively subtract the common difference.

 

Sn = an + (an – d) + (an – 2d) + … + [an – (n–1)d].

 

So, the sum of the arithmetic sequence could be found in any of the two ways. However, 

on adding those two equations together, we will be getting

 

Sn = a1 + (a1 + d) + (a1 + 2d) + … + [a1 + (n–1)d]

Sn = an + (an – d) + (an – 2d) + … + [an – (n–1)d]

_________________________________________

2Sn = (a1 + an) + (a1 + an) + (a1 + an) + … + [a1 + an]

____________________________________________

 

We can notice that all the terms that have d are added out. Hence,

2Sn = n (a1 + an)

Sn = [n(a1 + an)]/2

 

By substituting an = a1 + (n – 1)d into the last formula, we have

Sn = n/2 [a1 + a1 + (n – 1)d] ...Simplifying

Sn = n/2 [2a1 + (n – 1)d].

 

These two formulas help us to find the sum of an arithmetic sequence quickly.

FAQs on RD Sharma Class 10 Solutions Chapter 9 - Exercise 9.3

1. What are some of the most important formulae for arithmetic progressions?

Some of the most useful formulae in the topic arithmetic progressions are 


  • Formula of the Common difference of an AP: d = a2 - a1

  • nth term formula of an AP: an = a + (n - 1)d

  • Formula for the Sum of n terms of an AP: Sn = n/2(2a+(n-1)d)

2. How to find the number of terms of an AP?

The number of terms in an AP sequence can simply be found by the division of the difference between the last and the first terms by the common difference, and then adding 1 to it.

3. What is the difference between arithmetic progression and arithmetic sequence?

Arithmetic Sequence also known as Arithmetic Series is the sum of the elements of Arithmetic Progression. Arithmetic Progression is a number of sequences within any range that can give a common difference d.

4. What are the different types of progressions in mathematics?

There are three different types of progressions in mathematics. They are


  • Arithmetic Progression (AP)

  • Geometric Progression (GP)

  • Harmonic Progression (HP)

5. What is an arithmetic progression?

An arithmetic progression, abbreviated to AP, is a sequence in which the difference between every two terms which are consecutive is the same. In other words, we can say that an Arithmetic progression is a sequence in which all the terms except the first term are obtained by adding a specific number to its previous number.