NCERT Class 11-Math՚s: Chapter – 4 Principle of Mathematical Induction Part 5 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Question 15:

State whether the following proof (by mathematical induction) is true or false for the statement.

Proof: By the Principle of Mathematical induction, is true for ,

Again for some . Now we prove that

Answer: False

Since in the inductive step both the inductive hypothesis and what is to be proved are wrong.

4.3 EXERCISE

Short Answer Type

Question 1:

Give an example of a statement which is true for all but and are not true. Justify your answer.

Answer:

Question 2:

Give an example of a statement which is true for all . Justify your answer. Prove each of the statements in Exercises by the Principle of Mathematical Induction:

Answer:

Question 3:

is divisible by , for each natural number .

Answer:

is divisible by

Question 4:

is divisible by 7, for all natural numbers n.

Answer:

Let is divisible by 7;

Question 5:

is divisible by , for all natural numbers .

Answer:

Let is divisible by .

Question 6:

is divisible by 8, for all natural numbers n.

Answer:

is divisible by .

Question 7:

For any natural number n, is divisible .

Answer:

is divisible by .

Question 8:

For any natural number n, is divisible by , where x and y are any integers with .

Answer:

is divisible by

Question 9:

is divisible by , for each natural number .

Answer:

is divisible by ; .

Question 10:

is divisible by 6, for each natural number n.

Answer:

is divisible by

Question 11:

for all natural numbers .

Answer:

is true

Question 12:

for all natural number n.

Answer:

is true

Question 13:

, for all natural numbers .

Answer:

is true

Question 14:

for all natural numbers .

Answer:

is true;

Question 15:

for all natural numbers n.

Answer:

is true;