Let P(n) be the statement "`n^2-n+41`" is prime. Prove that P(1), P(2) and P(3) are true. Also prove that P(41) is not true. How does this not contradict the Principle of Induction ?

Let P(n) be the statement : “the arithmetic mean of n and (n + 2) is the same as their geometric mean”’. Prove that P(1) is not true. Also prove that if P(n) is true, then P(n + 1) is also true. How does this contradict the principle of Induction ?

Let P (n) denote the statement : “ 2^n gen ! ". Show that P(1), P(2) and P(3) are true but P(4) is not true.

Let P(n) be the statement ''2^n > 1’’ . Is P(1) true ?

Let P(n) be the statement ‘‘4^n > n’’ . If P(r) is true, prove that P(r +1) is also true.

If P (n) is the statement : "n (n + 1) (n + 2) is divisible by 12”, prove that P(3) and P(4) are true, but P(5) is not true.

If P (n) is the statement : "2^(3n)- 1 is an integral multiple of 7”, prove that P(1), P(2) and P (3) are true.

Let P (n) be the statement : “P(n) : 10n + 3 is prime’’. Is P(3) true ?

Let P (n) be the statement ''3^n >n'' . Is P (1) true?

If P(n) is the statement “n(n + 1) is even’’, then what is P(4) ?

Let P (n) be the statement ''3^n >n'' . If P (n) is true, prove that P(n + 1) is true.