Prove: if n is a natural number, then 2^2n - 1 must be divisible by 3.

Let P (n) be the statement.

Note 2^2 - 1 = 3, divisible by 3

Hence P (1) is true.

Assume P (k) is true, that is, there exists integer Q such that 2^ (2k) - 1 = 3Q

Then 2^[2 (k+1) ] - 1

=[2^ (2k) ] (4) - 1

= (3Q+1) (4) - 1

=12Q+3

=3 (4Q+1)

Since (4Q+1) is an integer, 2^[2 (k+1) ] - 1 is divisible by 3, that is, P (k+1) is true.

By the principle of mathematical induction, P (n) is true.