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

1.	prove that `3^(2n)-1` is divisible by 8, for all natural numbers n.
Answer» 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` `=83^(2k)+3^(2k)-1` `=83^(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.

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