# NCERT Class 11-Math's: Chapter –4 Principle of Mathematical Induction Part 3

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**Question 8:**

Prove by induction that for all natural number

**Answer:**

Consider

We observe that

is true, since

Assume that is true for some natural numbers , 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 number .

**Question 9:**

Prove by the Principle of Mathematical Induction that

for all natural numbers *n*.

**Answer:**

Note that is true, since

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

To prove is true, we have

Thus is true, whenever is true. Therefore, by the Principle of Mathematical Induction, is true for all natural number *n*.