Principle of Ma...

Let $$f\left ( n \right ) = 8^{n}-3^{n}$$, if $$n$$ is odd natural number then $$f\left ( n \right )$$ is divisible by

If $$\displaystyle P\left ( n \right )= 1^{2}+3^{2}+5^{2}+...+\left ( 2n-1 \right )^{2}=  \frac{n\left ( 4n^{2} -1\right )}{3}$$, then which of the following does NOT hold good?

If $$P\left ( n \right )= 1+2+3+\dots+n$$ is a perfect square, $$N$$ is less than $$100$$, then possible values of $$n$$ is/are

Let the statement $$P\left ( n \right ):2^{n}\geq 3n$$, the truth of $$P\left( k \right), \forall   k\in N  \Rightarrow$$ truth of,

Let $$f\left ( n \right )$$ equals to the sum of the cubes of three consecutive natural numbers.
$$f\left ( n \right )$$ leaves the remainder zero when divided by

The number of values of $$n$$, for which $$\displaystyle p(n)=1!\:+2!\:+3!\:+4!\:+\dots+\:n!$$ is the square of a natural number, is equal to

If $$n\in N$$, then $$n^{3}+2n$$ is divisible by

If $$\displaystyle P\left ( n \right )=\frac{n^{5}}{5}+\frac{n^{3}}{3}+\frac{7n}{15}$$, then statement "$$P(n)$$ is a natural number" is true,

Let $$P\left ( n \right )= n^{3}-n$$, the largest number by which $$P\left ( n \right )$$ is divisible $$\forall $$ possible integral values of $$n$$ is

For every natural number n -