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.