Is `2^(3-n)` a prime number for all natural numbers n?

Prove the statement by the principle of mathematical induction : 2^(3n) - 1 is divisible by 7, for all natural numbers n .

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

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

Write the negation of each of the following statements : (n^(2)+n+41) is a prime number for all n in A .

Prove the statement by the principle of mathematical induction : 3^(2n) - 1 is divisible by 8, for all natural number n.

prove that 2nlt(n+2)! for all natural numbers n.

prove that 2nlt(n+2)! for all natural numbers n.

P(n) is the statement : "n^2-n+7 is a prime number"(ii)Can you say that P(n) will be true for all natural numbers?give reason.

Use induction to prove that n^(3) - n + 3 , is divisible by 3, for all natural numbers n