Prove the statement by the Principle of Mathematical Induction :32n

1.	Prove the statement by the Principle of Mathematical Induction :32n – 1 is divisible by 8, for all natural numbers n.
Answer» According to the question, P(n) = 3²ⁿ – 1 is divisible by 8. So, substituting different values for n, we get, P(0) = 3⁰ – 1 = 0 which is divisible by 8. P(1) = 3² – 1 = 8 which is divisible by 8. P(2) = 3⁴ – 1 = 80 which is divisible by 8. P(3) = 3⁶ – 1 = 728 which is divisible by 8. Let P(k) = 3^2k – 1 be divisible by 8 So, we get, ⇒ 3^2k – 1 = 8x. Now, we also get that, ⇒ P(k+1) = 3^2(k+1) – 1 = 3²(8x + 1) – 1 = 72x + 8 is divisible by 8. ⇒ P(k+1) is true when P(k) is true. Therefore, by Mathematical Induction, P(n) = 3²ⁿ – 1 is divisible by 8, for all natural numbers n.

Prove the statement by the Principle of Mathematical Induction :32n – 1 is divisible by 8, for all natural numbers n.

Answer»

According to the question,

P(n) = 3²ⁿ – 1 is divisible by 8.

So, substituting different values for n, we get,

P(0) = 3⁰ – 1 = 0 which is divisible by 8.

P(1) = 3² – 1 = 8 which is divisible by 8.

P(2) = 3⁴ – 1 = 80 which is divisible by 8.

P(3) = 3⁶ – 1 = 728 which is divisible by 8.

Let P(k) = 3^2k – 1 be divisible by 8

So, we get,

⇒ 3^2k – 1 = 8x.

Now, we also get that,

⇒ P(k+1) = 3^2(k+1) – 1

= 3²(8x + 1) – 1

= 72x + 8 is divisible by 8.

⇒ P(k+1) is true when P(k) is true.

Therefore, by Mathematical Induction,

P(n) = 3²ⁿ – 1 is divisible by 8, for all natural numbers n.