Question

Using the principle of mathematical induction, prove that `(2^(3n)-1)` is divisible by `7` for all `n in Ndot`

Answer 1

`2^(3n) - 1 = (2^(3))^(n) - 1 = (1+7)^(n) - 1`
`= [1+.^(n)C_(1)(7) + .^(n)C_(2)(7)^(2) + "……" + .^(n)C_(n) (7)^(n)] - 1`
`= 7 [.^(n)C_(1) + .^(n)C_(2) (7) + "….." + .^(n)C_(n) (7)^(n-1)]`
Thus, `2^(3n) - 1` is divisible by 7 for all `n in N`