prove that `3^(2n)-1` is divisible by 8, for all natural numbers n.

Explanation

Let `P(n):3^(2n)-1` is divisible by 8, for all natural numbers.
Step I We observe that P(1) is true.
`P(1):3^(2(1))-1=3^(2)-1`
=9-1=8,which is divisible by 8.
Step II Now, assume that P(n) is true for n=k.
`P(k):3^(2k)-1=8q`
Step III Now, to prove P(k+1) is true.
`P(k+1):3^(2(k+1))-1`
`=3^(2k)*3^(2)-1`
`=3^(2k)*(8+1)-1`
`=8*3^(2k)+3^(2k)-1`
`=8*3^(2k)+8q`
`=8(3^(2k)+q)` [from step II]
Hence, 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.