The first part of the chapter aims at giving a basic idea of deductive reasoning. The questions asked from this part comprises two or three-line statement that the students are asked to prove true or false with the help of derived deductive steps.  

If you want to gain a clearer understanding of the theorems involved in deriving the deductive steps, you can download the NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction PDF.

In this section of Principles of Mathematical Induction Class 11, students get to learn about an important concept that is very crucial before solving the sums of mathematical induction and also includes  the real-life scenarios to make students understand the concept. The principle of motivation includes the process of proving that if a given statement is true for one natural number then, it also gets true for the rest of n natural numbers. It is important to develop a strong grip on this principle as it forms the very basis for the first principle of mathematical induction and also helps in solving related sums and problems.

This section teaches students about the process of deriving an equation. Students often struggle with this, so, The NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction prove to be of great help to students.

To get the full and final touch up in the chapter, NCERT Solutions for Mathematical Induction will guide you to better understand the principle of mathematical induction with the help of previously discussed concepts. You will be required to solve the equations based on the same.

Prove the following by using the principle of mathematical induction for all n ∈ N:

1.

Solution:

Therefore, P (k + 1) is true whenever P (k) is true.

Hence,  by the principle of mathematical induction, statement P (n) holds true for all natural numbers, i.e. n.

2. 

Solution:

Therefore, P (k + 1) is true whenever P (k) is true.

Hence, by the principle of mathematical induction, statement P (n) holds true for all natural numbers, i.e. n.

3.

Solution :

Therefore, P (k + 1) is true whenever P (k) is true.

Hence, by the principle of mathematical induction, statement P (n) holds true for all natural numbers, i.e. n

4.

Solution :

Therefore, P (k + 1) is true whenever P (k) is true.

Hence, by the principle of mathematical induction, statement P (n) holds true for all natural numbers, i.e. n.

5.

Solution :

Therefore, 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, i.e. n.

6.

Solution :

Therefore, 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, i.e. n.

7.

Solution:

Therefore, 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, i.e. n.

8.

Solution:

Therefore,  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, i.e. n.

 

• Explain the first principle of Mathematical Induction.

 

Answer: The first principle of Mathematical Induction states that when the base step and the inductive step are true, it can be concluded that the initial statement is true for all the natural numbers.

For a better understanding, you can find an ample number of questions based on this principle in CBSE Class 11 Maths Principle of Mathematical Induction Principle of Mathematical Induction (Make it bold).

 

• What do you mean by the concept of Mathematical Induction as stated in Class 11 Maths Chapter 4? 

 

Answer: Mathematical Induction is a method of confirming statements to prove a theorem, formula or statement which is true for all the given natural numbers. 

There are two steps to prove a statement: 

1. Base Step: in this, you will have to prove that a given statement P(n) is true for the initial value of n = 1. 

2. Inductive Step: This step is divided into further two steps-

1. 1. Suppose the statement P(n) is true for a value, say k, provided k is a positive integer.
   2. Now, use the previous assumption to prove that the given statement is true for n = k + 1, where k is a positive integer. 

After satisfying all the conditions, derive the conclusion that it is hence proved, statement P(n) holds true for all the natural numbers. 

 

• What is the importance of studying the Principles of Mathematical Induction? 

 

Answer: After studying the Principle of Mathematical Deduction, students understand the process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of the real numbers.

 

• What are the important characteristics of NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction ?

 

Answer: TheNCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction aim to provide a clear idea about the induction and deduction techniques that are used to prove theorems and equations. One significant basis for mathematical thinking is deductive reasoning. Contrary to deduction, inductive reasoning depends upon working with different cases and developing an assumption through observing the incidences until we have observed every case. The Principle of Mathematical Induction is one such tool that is used to prove numerous mathematical statements. Every given statement is assumed as P(n) related to a positive integer, say n, for which the validity of n = 1 is examined.

 

• How to use the Principle of Mathematical Induction? 

 

Answer: Mathematical Induction is a method used to check the validity of any theorem, formula or statement for a given natural number ‘n’. Firstly, check the validity of the first term i.e P(1). If this holds true, then we assume it holds its validity for the next integer as well. So, you need not check the validity of each integer.  Authentication of a specific case manifests the statement for the entire general case.

 

• What can you learn from the principles of mathematical induction ncert solutions?

 

Answer: The NCERT Solutions for Class 11 Maths Chapter 4 Principle of Mathematical Induction provides a detailed guide about the induction and deduction techniques used to prove theorems and statements. Students also get a thorough knowledge of the principles of mathematical deduction and its widely used applications through solving the numerous problems given on the topic. 

Our carefully curated solutions formulated by the subject experts will help you prepare for any question that you might encounter in the exams.

 

Is mathematical induction tough in class 11?

The initiation of the inductive step is the heart of this chapter, this might be one of the reasons students find mathematical induction a little trickier. The statement which is to be proven by the students is difficult to visualise because of its algebraic nature. This is especially accurate when it comes to trigonometric or algebraic statements. Therefore, it is very important to brush up on the concepts well and practice hard for the exams.

 

What is the concept of Mathematical Induction?

Mathematical Induction is basically a technique that is used to first validate any given theorem, statement or specific case and then arrive at a conclusion or assumption that it holds true for the given case and for the given natural numbers. It acts as a ripple effect where if a statement is proven true that k = 1, then it will hold true for all the natural numbers given. 

There are two steps involved in proving a statement which are- 

1. Base step

2. Inductive step

 

What is the use of Mathematical Induction in real-life Class 11? 

 

Structural Induction, an extension of Mathematical Induction, is used in mathematical logic, graph theory, computer science and some other mathematical fields to prove claims about more generic well-founded structures, such as trees. This good ordering theory is logically analogous to the reasonableness of mathematical induction. 

So, you must practice and solve all the questions and the related themes and concepts to hold a grip on the chapter. 

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