Prove the statement by the Principle of Mathematical Induction :23n

1.	Prove the statement by the Principle of Mathematical Induction :23n – 1 is divisible by 7, for all natural numbers n.
Answer» According to the question, P(n) = 2³ⁿ – 1 is divisible by 7. So, substituting different values for n, we get, P(0) = 2⁰ – 1 = 0 which is divisible by 7. P(1) = 2³ – 1 = 7 which is divisible by 7. P(2) = 2⁶ – 1 = 63 which is divisible by 7. P(3) = 2⁹ – 1 = 512 which is divisible by 7. Let P(k) = 2^3k – 1 be divisible by 7 So, we get, ⇒ 2^3k – 1 = 7x. Now, we also get that, ⇒ P(k+1) = 2^3(k+1) – 1 = 2³(7x + 1) – 1 = 56x + 7 = 7(8x + 1) is divisible by 7. ⇒ P(k+1) is true when P(k) is true. Therefore, by Mathematical Induction, P(n) = 2³ⁿ – 1 is divisible by7, for all natural numbers n.

Prove the statement by the Principle of Mathematical Induction :23n – 1 is divisible by 7, for all natural numbers n.

Answer»

According to the question,

P(n) = 2³ⁿ – 1 is divisible by 7.

So, substituting different values for n, we get,

P(0) = 2⁰ – 1 = 0 which is divisible by 7.

P(1) = 2³ – 1 = 7 which is divisible by 7.

P(2) = 2⁶ – 1 = 63 which is divisible by 7.

P(3) = 2⁹ – 1 = 512 which is divisible by 7.

Let P(k) = 2^3k – 1 be divisible by 7

So, we get,

⇒ 2^3k – 1 = 7x.

Now, we also get that,

⇒ P(k+1) = 2^3(k+1) – 1

= 2³(7x + 1) – 1

= 56x + 7

= 7(8x + 1) is divisible by 7.

⇒ P(k+1) is true when P(k) is true.

Therefore, by Mathematical Induction,

P(n) = 2³ⁿ – 1 is divisible by7, for all natural numbers n.