23n – 1 is divisible by 7 for all n>=1

n(1)

23(1) – 1 = (2*2*2) – 1

= 7

23n – 1 = 7m

n(k+1)

23(k+1) – 1 = 2.23k-1

= 2 * (7m+1) – 1

= 14m + 2 -1

= 14m-1

=7(2m-1)

Where (2m-1) is an int.

So 23n – 1 is divisible by 7.

Therefore, by P.M.I 23n – 1 is divisible by 7 for n>=1.