Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions

Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-8. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-8 (a), 8 (b). Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions

Class: 11th
Subject: Mathematics
Chapter  : Ch-8 Mathematical Induction of Section -A
Board ISC
Writer  OP Malhotra
Publications S.Chand Publications 2020-21

-: Select Topics :- 

Exe-8 (a)

Exe-8 (b)


Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions

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.

Let us take an example of the following pattern:

1=1²=1

4=2²=1+3

9==1+3+5

16==1+3+5+7

Here we can witness the sum of the first two odd natural numbers is square of the second number which is a natural number and the pattern continues.

P(n)=1+3+5+7+11..+(2n1)=n²

And P(1) is satisfied, then, it is the first step and the value will satisfy for all natural numbers.

Principle of Mathematical Induction

Consider a statement P(n), where n is a natural number. Then to determine the validity of P(n) for every n, use the following principle:

Step 1:  Check whether the given statement is true for n = 1.

Step 2: Assume that given statement P(n) is also true for n = k, where k is any positive integer.

Step 3:  Prove that the result is true for P(k+1) for any positive integer k.

If the above-mentioned conditions are satisfied, then it can be concluded that P(n) is true for all n natural numbers.


Exe-8 (a)

Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions

Question 1 :

1 + 2 + 3 +…. + n = 1/2 n (n+ 1)

Question 2 :

2  + 4 ………………………. (n +1).

Question 3 :

1² + ………………. (2n +1).

Question 4 :

1² + …………………… (2n +1).

Question 5 :

…………………………

………………………….

………………………….

Question 11 :

5 + 15 + 45 ……………………….. (3n-1 ).

Question 12 :

…………………….

………………………

…………………….

Question 17 :

Let S (k)= 1 + 3  + 5 + ………………… + (2k – 1 ) = 3 + k² , then which of the following is true?


Page 8-7 to 8-8Exe-8 (b)

Mathematical Induction OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 8-13 to 8-14

Question 1 :

n(n+ 1) (n + 5) is a multiple of 6. 

Question 2 :

n³ (n+1)³ ………………….. of 9

Question 3 :

……………………….

…………………….

……………………….

Question 9 :

Prove by the method of ……………………………. for all n ∈ N.

Question 11 :

(2n+7)……………….. natural numbers.

Question 12 :

1² + 2² + 3² +…………. + n² >  ……………. n ∈ N.

-: End of Mathematical Induction Solution :-

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