32n – 1 is divisible by 8,

Solution:

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) = 32k – 1 be divisible by 8

So, we get,

⇒ 32k – 1 = 8x.

Now, we also get that,

⇒  P(k+1) = 32(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.