Question

( 2³n - 1 )is divisiblw by( n € N )​

Answer 1

Answer:

Let P(n): 23n - 1 is divisible by 7

Now, P( 1): 23 - 1 = 7, which is divisible by 7.

Hence, P(l) is true.

Let us assume that P(n) is true for some natural number n = k.

P(k): 23k – 1 is divisible by 7.

or 23k -1 = 7m, m∈ N (i)

Now, we have to prove that P(k + 1) is true.

P(k+ 1): 23(k+1)– 1

= 23k.23 -1

= 8(7 m + 1) – 1

= 56 m + 7

= 7(8m + 1), which is divisible by 7.

Thus, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for all natural numbers n.

Hope it helps you