Revision Notes: Principle of Mathematical Induction

# Principle of Mathematical Induction Class 11 Notes Chemistry Chapter 4

``` Page 1

KEY POINTS
? A meaningful sentence which can be judged to be either true or
false is called a statement.
? A statement involving mathematical relations is called as
mathematical statement.
? Induction and deduction are two basic processes of reasoning.
? Deduction is the application of a general case to a particular
case. In contrast to deduction, induction is process of reasoning
from particular to general.
? Induction being with observations. From observations we arrive
at tentative conclusions called conjectures. The process of
induction help in proving the conjectures which may be true.
? Statements like(i)
( 1)
1 2 3 .........
2
n n
n
?
? ? ? ? ?
? n ? N.
(ii) 2 2
n
? ? n ? N.
(iii) If n(A)=n then number of all subsets of 2
n
A ? ? n ? N.
Page 2

KEY POINTS
? A meaningful sentence which can be judged to be either true or
false is called a statement.
? A statement involving mathematical relations is called as
mathematical statement.
? Induction and deduction are two basic processes of reasoning.
? Deduction is the application of a general case to a particular
case. In contrast to deduction, induction is process of reasoning
from particular to general.
? Induction being with observations. From observations we arrive
at tentative conclusions called conjectures. The process of
induction help in proving the conjectures which may be true.
? Statements like(i)
( 1)
1 2 3 .........
2
n n
n
?
? ? ? ? ?
? n ? N.
(ii) 2 2
n
? ? n ? N.
(iii) If n(A)=n then number of all subsets of 2
n
A ? ? n ? N.

(iv)
2( 1 )
1
n
r
Sn
r
?
?
?
where Sn is sum of n terms of G.P,  a = 1
st
term and r = common ratio. Are all concerned with n ? N which
takes values 1,2, 3,…. Such statements are denoted by P(n). By
giving particular values ton, we get particular statement as P(1),
P(2),…….. P(k) for some k ?N.
Principle of mathematical Induction:
Let P(n) be any statement involving natural number n such that
(i) P(1) is true, and
(ii) If P(k) is true ?P(k+1) is true for some some k ?N. that is
P(KH) is true whenever P(K) is true for some k ?N then
P(n) is true ? n ? N.
```

## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Principle of Mathematical Induction Class 11 Notes Chemistry Chapter 4

 1. What is the principle of mathematical induction?
Ans. The principle of mathematical induction is a proof technique used to establish that a certain statement is true for all natural numbers. It consists of two steps: the base step, where the statement is verified for the smallest natural number, and the induction step, where it is shown that assuming the statement holds for a particular natural number, it also holds for the next natural number.
 2. How is the principle of mathematical induction applied in JEE exams?
Ans. The principle of mathematical induction is often used in JEE exams to prove various mathematical statements involving natural numbers. Questions may require students to apply the principle to prove a given mathematical property or to solve problems that involve sequences or series. Understanding and practicing the application of mathematical induction is crucial for success in JEE exams.
 3. Can you provide an example of a JEE-level question that uses the principle of mathematical induction?
Ans. Sure! Here's an example: "Prove that for every positive integer n, the sum of the first n odd numbers is equal to n^2." To solve this question, one can use the principle of mathematical induction by verifying the base step (n = 1) and then proving the induction step, assuming the statement holds for a particular value of n and showing that it holds for (n + 1) as well.
 4. Are there any common mistakes students make when applying the principle of mathematical induction in JEE exams?
Ans. Yes, there are a few common mistakes that students make when applying the principle of mathematical induction. One mistake is assuming the statement is true for the induction step without properly verifying the base step. Another mistake is incorrectly applying the induction hypothesis, which may lead to an incorrect conclusion. It is important to carefully follow the steps of mathematical induction and double-check the reasoning at each step to avoid these errors.
 5. How can I improve my understanding and application of the principle of mathematical induction for JEE exams?
Ans. To improve your understanding and application of the principle of mathematical induction for JEE exams, it is recommended to practice solving a variety of problems that involve mathematical induction. Start with simpler examples and gradually move on to more complex ones. Additionally, studying the theory behind mathematical induction and understanding the logical reasoning involved will help you develop a strong foundation. Practice, practice, and practice some more to enhance your skills in applying the principle of mathematical induction.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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