ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding Chapter-4. Step by step Solutions of ML Aggarwal ISC Class-11 Mathematics with Exe1.1,  Exe-1.2, Exe-1.3, and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-11 Mathematics.

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding Chapter-4.

Class:	11th
Subject:	Mathematics
Chapter  :	Ch-4 Principle of Mathematical Induction of Section -A

Board	ISC
Writer	ML Aggarwal
Publications	APC Arya Publications 2020-21

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-: Select Topics :- 

Exe-4.1,

 Exe-4.2,

 Exe-4.3,

Chapter Test

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ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

What is Mathematical Induction?

Mathematical induction is a specialized form of working on different cases and coming up with observations. Induction is the compilation from a particular set of facts. This method is used to determine a wide range of statements in which we analyze the legitness of the case. The set should be denumerable in order for mathematical induction to work with an infinite set, meaning that it should be having a one-to-one correspondence between the elements of the set in question and the set of positive integers.

Properties of Mathematical Induction

Mathematical induction has to follow statements with respect to the properties they obey:

• When for the value of n is true for statements such as n ≥ 5, we should initiate with satisfying the value n=5 i.e. P(5)
• If the statement provided is true for n=k, and if it satisfies the value n=k then it will also satisfy n=k+1. In order to prove the trueness of the statement, we have to prove n=k+1.

Inductive Hypothesis

The step mentioned above is the assumption of the trueness of the statement n=k and is referred to as the inductive step or inductive hypothesis.

First Principle of Mathematical Induction

The proposition P (n) can be proved by the principle of mathematical induction for all n ∈ N as discussed in the following three steps:

1. The first step is also termed as the basic step of induction in which we prove that the position P(n) is true for the smallest positive integer or the first natural number i.e. for n = 1. This is also called as the verification step.

2. The next step is the induction step in which the proposition is assumed to be true for some n = k ≥ 1, i.e. we assume P(k) to be true.

3. The last step is the generalization step. If P(k) is true, then the proposition is proved true for n = k+1 i.e. for the next positive integer or the next natural number.

So, in this step the motive is to prove that if P(k) is true then P(k+1) must also be true, i.e. P(k) ⇒P(k+1) is true.

At the end, the result is generalized by saying that since the proposition is true for n = k+1, then it must be true for n = k as well and hence the proposition will also be true for all n belonging to the set of natural numbers.

Second Principle of Mathematical Induction

1. The first step is the verification step in which the proposition P(n) is proved to be true for n = m, where m is some fixed integer.

2. In the next step which is the induction step, we assume that the proposition P(n) is true for n = m, m+1, m+2, … r.

3. The last step is again the generalization step. In this step, prove that the proposition P(n) is true for n = r +1. Thus, once it is proved true, the result is generalized by stating that since the proposition is true for n = r+1, then it must also be true for n = m, m+1, m+2, … r as assumed in the previous step. Hence, the proposition is true for all n ≥ m belonging to the set of natural numbers.

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Exe-4.1,

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

Question 1:

If P (n) is the statement ……………………………….. then what is P (3) ?

Question 2:

If P (n) is the …………………………. is not true .

Question 3:

………………………

………………………

………………………

Question 6:

If P (n) denote the statement ………………………………. show thatm :

(i)……………..

(ii)……………..

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Exe-4.2

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

Using the principle of mathematical induction prove that for all n ∈ N :

Question 1:

2 + 4 + 6 +……………….

Question 2:

1 + 4 + 7 + …………….

Question 3:

…………………………

…………………………

…………………………

Question 23:

If x is not an integral multiple of π use induction to prove that

sin x + sin ……………

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 Exe-4.3,

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

Using the principle of mathematical induction prove that :

Question 1:

………………… all n ∈ N

Question 2:

2 ……………………. number n ≥2.

Question 3:

…………………..

…………………..

………………….

Question 5:

Let u1 = u2 = 5 ………………………. for all n ∈ N .

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Chapter Test

ML Aggarwal Principle of Mathematical Induction ISC Class-11 Maths Understanding

Question 1:

If (2.1 + 1) + …………………. that it is true for all n ∈ N .

Using the principle of mathematical induction prove that for all n ∈ N :

Question 2:

4 + 8+ 12 + …………… (n + 1)

Question 3:

……………………..

……………………..

……………………..

Question 10:

By induction , prove that :

…………………………..

Question 11:

By, induction, prove that n³ – n is divisible by 6, for all natural number n, n ≥ 2.

-: End of Principle of Mathematical Induction ML Aggarwal  ISC Class-11 Maths Chapter-4 Solution :-

Return to :-  ML Aggrawal ISC Class-11 APC Understanding Maths Solutions

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