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Mathematical Induction Class 11 Notes CBSE Maths Chapter 4 [Free PDF Download]

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Revision Notes for CBSE Class 11 Maths Chapter 4 (Principle of Mathematical Induction) - Free PDF Download

CBSE Class 11 Maths Notes Chapter 4 Principle of Mathematical Induction are offered by Vedantu online and is as per the latest syllabus. Our Mathematical Induction Notes are incorporated with shortcut techniques to help you solve numerical problems faster. As a result, you can also boost your exam scores with our Class 11 Maths Ch 4 Notes. Besides, you can also have the opportunity to build on your fundamentals from our Mathematical Induction Class 11 Notes which have detailed explanations to essential topics as well.


Download CBSE Class 11 Maths Revision Notes 2023-24 PDF

Also, check CBSE Class 11 Maths revision notes for all chapters:


Principle of Mathematical Induction Chapter-Related Important Study Materials
It is a curated compilation of relevant online resources that complement and expand upon the content covered in a specific chapter. Explore these links to access additional readings, explanatory videos, practice exercises, and other valuable materials that enhance your understanding of the chapter's subject matter.

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Principle of Mathematical InductionClass 11 Notes Maths - Basic Subjective Questions


Section–A (1 Mark Questions)

1. If P (n) is the statement “n (n + 1) is even”, then what is P (3).

Ans. Given, $P(n) ; n(n+1)$ is even,

So, $P(3)=3(3+1)=3(4)=12$

Hence, $P(3)=12, P(3)$ is also even.


2. If P (n) is the statement “n2 – n + 41 is prime” then P (2) is _________.

Ans. Given, $P(n) ; n^2-n+41$ is prime.

$$ \begin{aligned} & P(2)=2^2-2+41 \\ & =4-2+41=43 \end{aligned} $$


3. If xn -1 is divisible by x - k, then the least positive integral value of k is ______.

Ans. Let $P(n)=x^n-1$ is divisible by $x-k$. $P(1)=x-1$ is divisible by $x-k$.

Since $k=1$ is the possible least integral value of $k$.


4. If P (n): n(n + 1)(n + 5) is divisible by 6, then check the validity of P (2).

Ans. $P(n): n(n+1)(n+5)$ is divisible by 6 .

For $n=2, P(2)=2(2+1)(2+5)$

$$ =2 \times 3 \times 7=42 $$

which is divisible by 6

Hence, $P(2)$ is true.


5. For each $n\epsilon N,3^{2n}-1$ is divisible by_______.

Ans. For each $n \in N, \mathrm{P}(n): 3^{2 n}-1$

$$ \begin{aligned} & P(1)=8 \\ & P(2)=80=8 \times 10 \end{aligned} $$

Thus $P(n)$ is divisible by 8 .


Section–B (2 Marks Questions)

6. Let P (n) = 1 + 3 + 5 +........+ (2n - 1) = 3 + n2. Then, show that, P (k) is true implies that P (k+1) is true.

Ans. $P(1): 1=4$ is not true.

Let $P(k): 1+3+5+\ldots+(2 k-1)=3+k^2$ be true.

$$ \begin{aligned} & P(k+1)=1+3+5+\ldots+(2 k-1)+(2 k+1) \\ & =3+k^2+2 k+1=(k+1)^2+3 \\ & \therefore P(k) \text { is true } \Rightarrow P(k+1) \text { is true. } \end{aligned} $$

$\therefore P(k)$ is true $\Rightarrow P(k+1)$ is true.


7. Find the smallest positive integral value of n, which divides 32n - 2n + 1.

Ans. Putting $n=1$ in $3^{2 n}-2 n+1$ then, $3^{1 \times 2}-2 \times 1+1=9-2+1=8$, which is divisible by $2,4,8$.

Putting $n=2$ in $3^{2 n}-2 n+1$ then, $3^{2 \times 2}-2 \times 2+1=81-4+1=78$, which is divisible by 2 .

It is always divisible by 2 .


8. Find the natural number which always divides 72n + 23n-3 . 3n-1.

Ans. Putting $\mathrm{n}=1$ in $7^{2 \mathrm{n}}+2^{3 \mathrm{n}-3} \cdot 3^{\mathrm{nn-1}}$ then, $7^{2 \times 1}+2^{3 \times 1-3} \cdot 3^{1-1}=7^2+2^0 \cdot 3^0$

$$ =49+1=50 $$ Also, $\mathrm{n}=2$, then 

$$\begin{aligned}7^{2 \times 2}+2^{3 \times 2-3} \cdot 3^{2-1} & =2401+24 \\& =2425\end{aligned}$$

From (i) and (ii), it is always divisible by 25 .


9. Let $P(n):2^{n}< (1\times 2\times 3\times \cdots \times n)$ . Then find the smallest positive integer for which $P(n)$ is true.

Ans. Since, $P(1): 2<1$ is false.

$P(2): 2^2<1 \times 2$ is false.

$P(3): 2^3<1 \times 2 \times 3$ is false.

$P(4): 2^4<1 \times 2 \times 3 \times 4$ is true.

$$ \therefore n=4 \text {. } $$


10. If P (n): ‘10n + 3 is a prime number’, then prove that P (1) and P (2) are true.

Ans. $P(n): 10 n+3$ is a prime number.

For $n=1, P(1)=10 \times 1+3=13$ is a prime number.

$\therefore P(1)$ is true.

For $n=2, P(2)=10 \times 2+3=23$ is a prime number.

$\therefore P(2)$ is true.


11. If $x^{n}-y^{n}$ is divisible by x + y when n is an even integer, then check the validity of the statement for n = 4.

Ans. Let $P(n): x^n-y^n$ is divisible by $\mathrm{x}+\mathrm{y}$.

For $n=4$,

$$ \begin{aligned} P(4)=x^4-y^4 & =\left(x^2\right)^2-\left(y^2\right)^2 \\ & =\left(x^2+y^2\right)\left(x^2-y^2\right) \\ & =\left(x^2+y^2\right)(x+y)(x-y) \end{aligned} $$

Which is divisible by $x+y$.

Hence, $P(4)$ is true.


12. If $P(n)=2.4^{2n+1}+3^{3n+1}$ is divisible by k for all $k\epsilon N$ is true, then find the value of k.

Ans. $$ \begin{aligned} P(1) & =2.4^{2+1}+3^{3+1}=2.4^3+3^4 \\ & =2(64)+81=128+81=209 \end{aligned} $$

For $n=2$,

$$ \begin{aligned} P(2) & =2.4^5+3^7=8(256)+2187 \\ & =2048+2187=4235 \end{aligned} $$

Note that the H.C.F. of 209 and 4235 is 11.

So, $2.4^{2 n+1}+3^{3 n+1}$ is divisible by 11 .

Hence, $\mathrm{k}=11$.


13. For $P(n): n^2>100,P(k)$ is true, then prove that $P(k+1)$ is also true.

Ans. $P(n): n^2>100$

$P(k)$ is true.

$$ \begin{aligned} & \Rightarrow k^2>100 \\ & \Rightarrow k^2+2 k+1>100+2 k+1 \\ & \Rightarrow(k+1)^2>100+2 k+1 \\ & \Rightarrow(k+1)^2>100 \\ & \Rightarrow P(k+1) \text { is true. } \end{aligned} $$


PDF Summary - Class 11 Maths Principle of Mathematical Induction Notes (Chapter 4)


1. Deduction: Generalization of Specific Instance

Example: Rohit is a man, and all men consume food, hence Rohit eats food.

2. Induction: Specific Instances to Generalization

Rohit, for example, eats. Vikash consumes food. Vikash and Rohit are both males. The statement All men consume food is true for \[n=1\], \[n=k\], and \[n=k+1\], and it is also true for all natural integers n. 


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1. Steps of Principle of Mathematical Induction:

Allow \[\text{P}\left( \text{n} \right)\] to be a result or statement expressed in terms of n. (given question).

Step 2: Demonstrate that \[\text{P}\left( \text{1} \right)\] is correct.

Step 3: Assume \[\text{P}\left( \text{k} \right)\] is correct.

Step 4: Using Step 3 as a guide, show that \[\text{P}\left( \text{k+1} \right)\] is correct.

Step 5: As a result, whenever \[\text{P}\left( \text{k} \right)\] is true, \[\text{P}\left( \text{1} \right)\] is true and \[\text{P}\left( \text{k+1} \right)\] is true.

As a result, \[\text{P}\left( \text{n} \right)\] is true for all natural integers n, according to the Principle of Mathematical Induction.

Example: Prove that \[{{\text{2}}^{\text{n}}}\text{  n}\]for all positive integers n

Solution:

Step 1: Let \[\text{P}\left( \text{n} \right)\text{: }{{\text{2}}^{\text{n}}}\text{  n}\]

Step 2: When \[\text{n =1, }{{\text{2}}^{\text{1}}}\text{ 1}\]. Hence \[\text{P}\left( \text{1} \right)\] is true. 

Step 3: Assume that \[\text{P}\left( \text{k} \right)\] is true for any positive integer k, i.e., \[{{\text{2}}^{\text{k}}}\text{  k }...\text{ }\left( \text{1} \right)\]

Step 4: We shall now prove that \[\text{P}\left( \text{k +1} \right)\] is true whenever \[\text{P}\left( \text{k} \right)\] is true. 

Multiplying both sides of (1) by 2, we get

Therefore, \[\text{P}\left( \text{k+1} \right)\] is true when \[\text{P}\left( \text{k} \right)\] is true. Hence, by principle of mathematical induction, \[\text{P}\left( \text{n} \right)\] is true for every positive integer n.

Class 11 Maths Chapter 4 Notes Mathematical Induction – In a Nutshell

Mathematical Induction is a specific technique that is primarily used to prove a given statement or a theorem. The crucial part of this method is that the theorem should stand true for every natural number. It is a useful method as you do not have to go round solving an equation or a statement for every possible value it can take.

With Mathematical Inductions, solving numerically related to series becomes convenient, and as a result, it has extensive scopes of applications in real life as well. Computer science is one of the most renowned spheres, where it is widely used. 

Consequently, appropriate guidance of the method is necessary right from school-level, to enable students to make the most of these methods in simplifying the solutions. 

It is the reason that our subject experts have drafted the Mathematical Induction Notes to help you understand the topic precisely. Revising from our notes before your exam will assist you in recalling the crucial topics that you might have forgotten or missed while studying the chapter.

Students need to be familiar with the following pointers when studying Mathematical Induction:

  • Proving a given statement or illustration is the primary motive.

  • The proof should stand true for all values consisting of natural numbers.

  • The statement should be true for the initial value as well.

  • It should be true for all other values till nth iteration.

  • Every step involved in the proof must be justified and true.

You should have the necessary reasoning and logical skills to ace the numerical related to this chapter. They should be aware of the properties of natural numbers too before delving into solving numerical. 

Principle of Mathematical Induction Class 11 – Revision Notes

Here is a brief description of the topics covered in this chapter which should be considered by students, when they revise our Mathematical Induction Notes.

(i) Mathematical Induction Class 11 Notes - Principle of Mathematical Induction

There are two principles of Mathematical Induction, which you need to know right from the beginning. They are:

  • Deduction

  • Induction

This chapter deals with the latter principle. As you go through our revision notes, you will know that the former principle is based on a generalisation of certain specific instances, while the latter is opposite of it. Induction is more like particular instances of generalisation. 

You will find a simple example in our notes of Principle of Mathematical Induction Class 11 to facilitate a more fundamental understanding of both the concepts. As a result, you will also be able to differentiate between the two. The given example is as follows:

Example: 

  1. Deduction: In here, you are provided with some facts or statements, out of which you are required to deduce or derive a particular statement or information. It is similar to deciphering the required information from the data. 

For instance, consider this example-

Statement 1: Rohit is a man.

Statement 2: All men eat food.

Conclusion: Rohit eats food.

As you can see in our Mathematical Induction Notes, the two statements above help in deriving the conclusion that says Rohit is one of the men who eats food.

Induction: Herein, you will be provided with statements that signify specific instances, out of which you are likely to draw conclusions related to generalisations.

Consider this example-

Statement 1: Rohit eats food.

Statement 2: Vikas eats food.

Statement 3: Rohit and Vikas are men.

Conclusion: All men eat food.

As can be seen, you can derive the conclusion out of the three given statements which are specific instances. Therefore, you can induce a generalised statement from here. Make sure you go through each line in our Mathematical Induction Notes to ensure you gain clarity on both the topics.

In addition, we have also presented a mathematical deduction for the same. It will help you understand the concepts in a more natural way. Since the main objective is to solve numerical with the help of this method, we have presented the same with the support of algebraic representations.

For instance,

If a statement is valid for a value of n, where n = 1,

And again, the same statement is true another value, say n = k,

Once again, it is valid for n = k +1,

Therefore, one statement is true for all the above-mentioned values; as a result, it is plausible that it will be valid for all other values of n, provided they are natural numbers. 

Refer to our Mathematical Induction Notes for a more logical explanation of the same. We have made a diagram with proper labelling that sums up the original idea behind induction and deduction.

ii. Class 11 Maths Chapter 4 Notes  - Steps in Mathematical Induction

There are five steps in all you need to follow while solving a numerical using mathematical induction. In our revision notes, the steps are explained in brief so that you can easily recall them and solve the numerical.

Students should be aware that these shortcut techniques are mainly for fulfilling the purpose of simplifying the process of memorising the method of Mathematical Induction. Hence, they need to have clarity on the topics in details at the first place after which they can refer to our revision notes before the exam.

Although the diagrammatic representation in the previous section can help them in understanding the concerned topic quickly, they should not miss out any pointer written in the notes. Each tip given in our Class 11 Maths Chapter 4 Notes is equally important and must be kept in mind to understand the topic in depth.

The steps involved in solving numerically related to this topic can be summarised as follows:

  • Step 1: Consider P(n) to be a given statement or resultant in terms of ‘n’.

  • Step 2: Prove P(1) to be true.

  • Step 3: Consider that P(k) is also true.

  • Step 4: Using the previous step, make sure P(k+1) is also true.

  • Step 5: Now, both P(1) and P(k+1) is valid.

Consequently, by the Principle of Mathematical Induction, P(n) is true for all values of natural numbers ‘n’. To ensure that students do not miss out on any significant step, we have presented the steps one after one briefly without compromising on the main content. 

Simultaneously, it will also help you to boost your exam scores. Revise these Mathematical Induction Notes regularly before your actual exam, and you can add an extra edge over to the competition. 

iii. Mathematical Induction Class 11 Notes - Illustrated Example 

To further simplify your understanding, we have presented an example too. Here the questions ask you to prove than 2n is greater than n for all positive integers that n can hold. As already mentioned, we follow the same steps one by one in the following manner:

  • Step 1: Let P(n): 2n > n

  • Step 2: When n =1, 21 = 2, which is greater than 1. Thus, P(1) is true.

  • Step 3: Assuming P(k) is valid for any natural number k, 2k > k.

  • Step 4: Now, you have to prove that P(k+1) is also true, as P(k) is true. Take note of the following steps given in our Mathematical Induction Notes.

We have the equation 2k > k, so we multiply each side by 2, 

Then we get 21. 2k > 2. k

or, 2(1+k) > 2k

or, 2(1+k) > k + k

or, 2(1+k) > k + 1 [since, k > 1]

Therefore, it can be seen that P(k+1) is true when P(k) is true. Again, by the Principle of Mathematical Induction, P(n) stands true for all values of n which re natural numbers. These steps are illustrated in our Principle of Mathematical Induction Class 11 Notes so that you can understand the topic clearly. 

These study-notes are standard and updated as per the latest syllabus so that you can cover every significant pointer included in the chapter. Therefore, make sure you go through the same and improve your chances of scoring higher in the exam.

Select Vedantu - A High-Class Knowledge Hub as your Learning Partner!

At Vedantu, we house some of the most efficient subject experts who have been teaching students for more than a decade now. They have mastered their respective subjects and have drafted the Principle of Mathematical Induction Class 11 Notes PDF with utmost care and perfection.

These revision notes have all the necessary details and explanation required to ace thee exams. You can go through these Mathematical Induction Notes and take note of the shortcut techniques and formulas listed in them which will, in turn, help you to solve the sums faster. Also, you can recall the critical pointers from here too.

FAQs on Mathematical Induction Class 11 Notes CBSE Maths Chapter 4 [Free PDF Download]

1. What are the principles of mathematical induction?

Deduction and Induction are the two principles of Mathematical Induction. In deduction, you are required to deduce or derive a particular statement or information from the facts or statements provided to you. Basically, it means generalization of specific instances. It is almost like deciphering the required information from the data. In Induction, you will be given statements that signify specific instances. From those statements, you are likely to draw conclusions related to generalisations.

2. Why are Revision Notes of Chapter 4 of Class 11 Maths important?

The Revision Notes of Class 11 Maths Chapter 4 cover all the important concepts related to mathematical induction and deduction. Each topic provided in these notes is designed as per the CBSE guidelines to make the students gain knowledge on the core concepts. The format of these notes is very well designed in such a way that students get a clear understanding of all the topics related to the principle of induction. These well-structured resources provided free of cost by Vedantu are very useful to increase the accuracy of answers. Students must refer to these notes regularly as it will help them to gain confidence to score high marks in exams.

3. How can I prepare for the Chapter 4-Principle of Mathematical Induction on my own?

The most crucial aspect of studying a subject like Mathematics is to be clear with the basics. You can, thus, work hard and work smartly to understand the basic concept of the chapter on your own. You can start by getting a basic idea of the chapter either through the videos available online or going through the chapter. Once you are familiar with the concept and the content of the chapter, read it thoroughly. Write down important concepts and formulas as well as practice the examples to understand the concept better. Practice the given exercises, learn from your mistakes and then practice again.

4. Is there any need to practice extra questions of Chapter 4 of Class 11 Maths and is 1 day enough to complete studying the entire chapter?

There is no need to practice questions of Chapter 4 (Principle of Mathematical Induction) of Class 11 Maths from resources other than the NCERT books. The NCERT textbook questions and Exemplar questions as well as examples from the book are highly sufficient to practice. It depends on the student to practice extra questions or not. It is not compulsory. One day isn’t enough for the students to complete the entire Chapter 4 of Class 11 Maths. A minimum of 3-4 days is required to complete this chapter if they spend 1-2 hours per day practising.

5. What are the important topics covered in Class 11 Maths Chapter 4 and how many questions are provided?

Basic terms and definitions related to the principle of mathematical induction are covered in these solutions. These solutions briefly introduce the students to deductive reasoning along with the principle of mathematical induction and motivation. To make the students easily relate to the concept, topics are explained with the help of examples. There are 24 questions provided in the NCERT book which will provide you with all the concepts required to study this topic.  To know more students can refer to the vedantu app.