Principle of Ma...

If $$P$$ is a prime number then $$n^{p} - n$$ is divisible by $$p$$ when $$n$$ is a 

The difference between an $$+$$ve integer and its cube, is divisible by

If n is a natural number then $$\left( \displaystyle \frac{n + 1}{2} \right)^{n}\, \geq n!$$ is true when.

For all n $$\in$$ N, $$n^{4}$$ is less than

A student was asked to prove a statement by induction. He proved
(i) P(5) is true and
(ii) Trutyh of P(n) $$\Rightarrow$$ truth of p(n + 1), n$$\in$$N
On the basis of this, he could conclude that P(n) is true for

For  every natural number $$n$$, $$n(n + 3)$$ is always :

For every positive integral value of n, $$3^n > n^3$$ when

$$P(n) : 3^{2n+2} -8n -9$$ is divisible by 64, is true for

The smallest positive integer for which the statement $$3^{n+1} < 4^n$$ holds is

For every positive integer $$n, \dfrac {n^7}{7}+\dfrac {n^5}{5}+\dfrac {2n^3}{3}-\dfrac {n}{105}$$ is