NCERT Class 11-Math՚S: Chapter – 4 Principle of Mathematical Induction Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 10:

Show by the Principle of Mathematical Induction that the sum of the term of the series is given by

Answer:

Also, note that any term of the series is given by

We observe that is true since

Assume that is true for some natural number k, i.e..

Case 1: When is odd, then is even. We have

So is true, whenever is true in the case when k is odd.

Case 2: When is even, then is odd.

Now,

Therefore, is true, whenever is true for the case when is even. Thus is true whenever is true for any natural numbers k. Hence, true for all natural numbers.

Objective Type Questions

Choose the Correct Answer in Examples 11 and 12 (M. C. Q.)

Question 11:

Let ″ . Then the smallest positive integer for which is true is

(A)

(B)

(C)

(D)

Answer:

Answer is D, since

is false

is false

is false

But is true

Question 12:

A student was asked to prove a statement by induction. He proved that is true whenever is true for all and also that is true. On the basis of this he could conclude that is true

(A)

(B)

(C)

(D)

Answer:

Answer is (C) , since is true and is true, whenever is true.

Fill in the blanks in Example 13 and 14.

Question 13:

If is divisible by for all ″ is true, then the value of is ________

Answer:

Now, for ,

for

Note that the H. C. F. of and is . So is divisible by .

Hence, is

Question 14:

If is divisible by for ″ is true, then the least negative integral value of is ________.

Answer:

For is divisible by .

Thus should be since, is divisible