NCERT Class 11-Math՚S: Chapter – 4 Principle of Mathematical Induction Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 4:

is divisible by 3.

Answer:

Let the statement given as

is divisible by , for every natural number .

We observe that is true, since

.

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

is divisible by , i.e.. , , where

Now, to prove that is true, we have

Thus is true, whenever is true.

Hence, by the Principle of Mathematical Induction is true for all natural numbers n.

Question 5:

, for all natural numbers .

Answer:

Let be the given statement, i.e.. , for all natural numbers, . We observe that is true, since

Assume that P (n) is true for some natural number

To prove is true, we have to show that Now, we have

Thus is true, whenever is true.

Hence, by the Principle of Mathematical Induction is true for all natural numbers, .

Long Answer Type

Question: 6

Define the sequence as follows:

, for all natural numbers .

(i) Write the first four terms of the sequence.

(ii) Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula for all natural numbers.

Answer:

(i) We have

(ii) Let be the statement, i.e.. ,

for all natural numbers. We observe that is true

Assume that P (n) is true for some natural number , i.e.. , .

Now to prove that is true, we have

Thus is true whenever is true.

Hence, by the Principle of Mathematical Induction, is true for all natural numbers.

Question 7:

The distributive law from algebra says that for all real numbers and , we have .

Use this law and mathematical induction to prove that, for all natural numbers, , if are any real numbers, then

Answer:

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

for all natural numbers for

We observe that P (2) is true since

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

Now to prove is true, we have

Thus is true, whenever is true.

Hence, by the principle of Mathematical Induction, is true for all natural numbers .