# NCERT Class 11-Math՚S: Chapter – 4 Principle of Mathematical Induction Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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;