# Chapter 4:(Principle Of Mathematical Induction)

Prove the following by using the principle of mathematical induction for all n ∈ N (Exercise 4.1 ncert math solution class 11)

Question 1:-

Solution: Let be the given statement

For ,

Assume that is true for some positive integer , such that

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 2:-

Solution: Let be the given statement.

For ,

which is true for n= 1

Let is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 3: .

Solution: Let be the given statement.

For ,

,

which is true for n= 1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Since

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 4:

Solution: Let be the given statement.

For ,

which is true.

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 5: .

Solution: Let be the given statement.

For ,

which is true

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 6: .

Solution: Let be the given statement. i.e.,

For ,

which is true for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 7: .

Solution: Let be the given statement. For ,

Hence, for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 8: .

Solution: Let be the given statement.

For ,

for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e.,

Question 9: .

Solution: Let be the given statement.

For ,

Hence, L.H.S. = R.H.S. for n= 1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 10:

Solution: Let be the given statement.

For ,

L.H.S.=R.H.S. for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 11: .

Solution: Let be the given statement.

For ,

Hence L.H.S.=R.H.S. for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 12: .

Solution: Let be the given statement.

For ,

Hence, L.H.S.=R.H.S. for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 13 : .

Solution: Let be the given statement.

For ,

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 14 :.

Solution: Let be the given statement.

For ,

Hence, L.H.S. = R.H.S. for n = 1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 15: .

Solution: Let be the given statement.

For ,

Hence, L.H.S. = R.H.S. for n=1

Assume that is true for some positive integer

We will now prove that is also true. Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 16: .

Solution: Let be the given statement.

For ,

Hence, L.H.S.=R.H.S., for n = 1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 17: .

Solution: Let be the given statement.

For ,

Hence, L.H.S. = R.H.S. for n=1

Assume that is true for some positive integer

We will now prove that is also true.

Now, we have

Taking L.H.S.

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 18 : .

Solution: Let be the given statement.

For ,

Since,

Hence, is true

Assume that is true for some positive integer

We will now prove that is true whenever is true Now, we have

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 19: is a multiple of 3.

Solution: Let the statement be P(n)

For

Thus is true for

Let be true for some natural number ,

Hence

where .

Now, we will prove that is true

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 20:  is divisible by 11 .

Solution: Let the statement be P(n)

For

Thus is true for

Let be true for some natural number ,

We can write

Now, we will prove that is true

Now,

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 21: is divisible by .

Solution: Let the statement be P(n)

For

Thus is true for

Let be true for some natural number ,

We can write it as

Now, we will prove that is true

Now,

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 22: is divisible by 8 .

Solution: Let the statement be P(n)

For

Thus is true for

Let be true for some natural number ,

where .

Now, we will prove that is true .

Now,

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 23: is a multiple of 27.

Solution: Let the statement be P(n)

Thus is true for

Let be true for some natural number ,

We can write

Now, we will prove that is true

Now,

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

Question 24 :

Solution:Let be the given statement.

We note that is true for ,

Assume that is true for some positive integer

We will now prove that is true,

Adding 2 both side in eq (i)

Thus is true, whenever is true.

Hence, from the principle of mathematical induction, the statement is true for all natural numbers i.e., .

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