NCERT Class 11-Math՚S: Chapter – 4 Principle of Mathematical Induction Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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.