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**NCERT Solutions for Class 11 Maths Chapter 4 – Principle of Mathematical Induction Exercise 4.1-** NCERT solutions for exercise 4.1 Class 11 Maths Chapter 4 deals with the principle of mathematical induction and related problems. Exercise 4.1 Class 11 Maths deals with how to prove a given mathematical statement using the principle of mathematical induction. NCERT Solutions for Class 11 Maths chapter 4 exercise 4.1 gives an insight into the steps of proving a given statement using the idea of induction. Solving both example questions and Class 11 Maths chapter 4 exercise 4.1 are important to understand the concepts discussed in the chapter. Before starting the Class 11 Maths chapter 4 exercise 4.1 the NCERT book discusses the concept of deduction in brief and introduces what is induction and how induction is different from deduction. Followed by examples and Class 11 Maths chapter 4 exercise 4.1.

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- NCERT Solutions for Class 11 Maths Chapter 4 - Principle of Mathematical Induction Exercise 4.1- Download Free PDF
- NCERT Solutions for Class 11 Maths Chapter 4 – Principle of Mathematical Induction Exercise 4.1
- Access Principle Of Mathematical Induction Class 11 Chapter 4- Exercise 4.1
- Question:1 Prove the following by using the principle of mathematical induction for all :
- More About NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1
- Key Features of 11th Class Maths Exercise 4.1 Answers
- NCERT Solutions of Class 11 Subject Wise
- Subject Wise NCERT Exampler Solutions

The NCERT Solutions for Class 11 Maths Exercise 4.1 are expertly crafted to help students overcome challenges in problem-solving. They provide step-by-step explanations and a free PDF version for convenient access, ensuring that students can grasp mathematical concepts effectively and excel in their studies.

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Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:2 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:3 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:4 Prove the following by using the principle of mathematical induction for all :

Answer:

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:5 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:6 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by principle of mathematical induction , statement p(n) is true for all natural numbers n

Question:7 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:8 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have , which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:9 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:10 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:11 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:12 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:13 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:14 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this statement is true

Now,

For n = k + 1 we have

&nbsnbsp;

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:15 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:16 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:17 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:18 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Question:19 Prove the following by using the principle of mathematical induction for all : is a multiple of

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is multiple of 3, hence true

For n = k we have

, Let's assume that this is multiple of 3 = 3m

Now,

For n = k + 1 we have

Where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is multiple of 3 for all natural numbers n

Question:20 Prove the following by using the principle of mathematical induction for all : is a divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 11, hence true

For n = k we have

, Let's assume that this is divisible by 11 = 11m

Now,

For n = k + 1 we have

nbsp;

Where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 11 for all natural numbers n

Question:21 Prove the following by using the principle of mathematical induction for all : is divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by , hence true

For n = k we have

, Let's assume that this is divisible by

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by for all natural numbers n

Question:22 Prove the following by using the principle of mathematical induction for all : is divisible by

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 8, hence true

For n = k we have

, Let's assume that this is divisible by 8 = 8m

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 8 for all natural numbers n

Question:23 Prove the following by using the principle of mathematical induction for all : is a multiple of

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is divisible by 27, hence true

For n = k we have

, Let's assume that this is divisible by 27 = 27m

Now,

For n = k + 1 we have

where some natural number

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is divisible by 27 for all natural numbers n

Question:24 Prove the following by using the principle of mathematical induction for all :

Answer:

Let the given statement be p(n) i.e.

For n = 1 we have

, which is true

For n = k we have

, Let's assume that this is true

Now,

For n = k + 1 we have

Thus, p(k+1) is true whenever p(k) is true

Hence, by the principle of mathematical induction, statement p(n) is true for all natural numbers n

Before the introduction of exercise 4.1 Class 11 Maths 8 example problems are given in NCERT chapter 4 of Class 11. Only one exercise is discussed in this chapter, which is Class 11 Maths chapter 4 exercise 4.1. 24 questions are discussed in the Class 11 Maths chapter 4 exercise 4.1. Solving all these questions are important as students may face similar or same questions in the exam paper. Give a good number of try to solve questions before looking to the NCERT solutions for Class 11 Maths chapter 4 exercise 4.1.

**Also Read| **Principle Of Mathematical Induction Class 11th Notes

**Benefits of NCERT Solutions for Class 11 Maths Chapter 4 Exercise 4.1**

Solving all the questions of exercise 4.1 Class 11 Maths gives a good idea of the steps involved in solving a given problem using mathematical induction

From this 11th class maths exercise 4.1 answers, students can definitely expect a question for the Class 11 final exam. To solve questions in the exam paper it is important to practice NCERT solutions for Class 11 Maths chapter 4 exercise 4.1.

**Step-by-step explanations**: Detailed ex 4.1 class 11 solutions with a step-by-step breakdown for each problem.**Clarity and precision**: Clear and accurate presentation, helping students confidently prepare for exams.**Conceptual understanding**: Emphasis on fostering a deep understanding of mathematical concepts.**Curriculum alignment**: class 11 maths ex 4.1 solutions closely follow the NCERT curriculum, covering topics as per the official syllabus.**Variety of practice problems**: A range of class 11 maths chapter 4 exercise 4.1 exercises provided for students to enhance their problem-solving skills.**Accessibility**: Class 11 ex 4.1 solutions are often available for free, making them easily accessible to students.**Format options**: PDF versions are typically offered for convenient online and offline access.

- NCERT Solutions for Class 11 Maths
- NCERT Solutions for Class 11 Physics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Biology

1. How many topics are given in the NCERT Class 11 chapter 4?

Three topics

2. What are the three topics given in the chapter principle of mathematical induction?

Introduction, motivation and the principle of mathematical induction

3. How many questions are solved in the Class 11 Maths chapter 4 exercise 4.1?

24 questions

4. What number of solved examples are given in the NCERT Class 11 Maths Chapter 4 - Principle of Mathematical Induction?

Eight examples are solved before the Exercise 4.1 Class 11 Maths

5. What is the relation between the principle of mathematical induction and Peano’s axioms?

It is a restatement of one of Peano’s axioms

6. Who was the first person to name and define mathematical induction?

De Morgan

7. Which mathematician is credited with the origin of the principle of mathematical induction?

Blaise Pascal

8. Why do we solve NCERT Solutions for Class 11 Maths chapter 4 exercise 4.1?

The basic aim of solving the NCERT exercise is to understand the concepts well and to get an insight in to different types of questions that uses the concepts studied. Other than this Class 11th Maths chapter 4 exercise 4.1 will help in exams also.

Application Date:20 November,2023 - 19 December,2023

Application Date:20 November,2023 - 19 December,2023

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A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

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Individuals who opt for a career as a talent director are professionals who work in the entertainment industry. He or she is responsible for finding out the right talent through auditions for films, theatre productions, or shows. A talented director possesses strong knowledge of computer software used in filmmaking, CGI and animation. A talent acquisition director keeps himself or herself updated on various technical aspects such as lighting, camera angles and shots.

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Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

2 Jobs Available

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

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In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

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Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

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Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available

3 Jobs Available

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

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**Quality Assurance Manager Job Description:** A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.

2 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available

Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.

2 Jobs Available

2 Jobs Available

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available

3 Jobs Available

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available

Big Data Analytics Engineer Job Description: A Big Data Analytics Engineer is responsible for collecting data from various sources. He or she has to sort the organised and chaotic data to find out patterns. The role of Big Data Engineer involves converting messy information into useful data that is clean, accurate and actionable.

2 Jobs Available

A Cloud Solutions Developer is basically a Software Engineer with specialisation in cloud computing. He or she possesses a solid understanding of cloud systems including their operations, deployment with security and efficiency with no little downtime.

2 Jobs Available

A Customer Relationship Management Technology Consultant or CRM Technology Consultant is responsible for monitoring and providing strategy for performance improvement with logged calls, performance metrics and revenue metrics. His or her role involves accessing data for team meetings, goal setting analytics as well as reporting to executives.

2 Jobs Available

Career as IT Manager requires managing the various aspects of an organization's information technology systems. He or she is responsible for increasing productivity and solving problems related to software and hardware. While this role is typically one of the lower-level positions within an organisation, it comes with responsibilities related to people and ownership of systems.

2 Jobs Available

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