Show that 2^2n-1 +1 is divisible by 3 for all n > 1.

Step-by-step explanation:

Let's assume that

is divisible by 3.

for n= 1

       = 3

Hence, P(1) is true for n=1

for n=2

       = 9, which is divisible by 3.

As we can see, P(n) is also true for n= 2.

Let's say P(n) is true for n = k

So,

is divisible by 3.

For P(n) should be true, the difference of P(k+1) and P(k) must will have divisible by 3.

So,

as P(k+1)-P(k) is divisible by 3.

As a result, P(n) is true for all n>1.