Principle of Ma...

If $$\displaystyle a_{n}=\sqrt{7+\sqrt{7+\sqrt{7+.....}}} $$ having $$n$$ radical signs, then by method of mathematical induction which of the following is true?

Let $$P(n)\, :\, n^{2}\, +\, n$$ is an odd integer. It is seen that truth of $$P(n)\, \Rightarrow$$ the truth of P(n + 1). Therefore, P(n) is true for all -

Let $$P\left ( n \right )= x^{2n-1}+y^{2n-1}$$ is divisible by $$x+y$$ as $$P\left ( 1 \right )$$ is true, then truth of $$P\left ( k+1 \right )$$ indicates

The sequence $$\displaystyle \left ( x_{n}n\geq 1 \right )$$ is defined by $$\displaystyle x_{1}=0 $$ and $$\displaystyle x_{n+1}=5x_{n}+\sqrt{24x^{2}_{n}+1} $$ for all $$\displaystyle n\geq 1.$$ Then all $$\displaystyle x_{n} $$ are 

The inequality $$n!\, >\, 2^{n\, -\, 1}$$ is true - 

Let $$P\left ( n \right )=11^{n+2}+12^{2n+1}$$, then the least value of the following which $$P\left ( n \right )$$ is divisible by is

For positive integer n, $$3^{n} < n!$$ when

Let $$P(n) : n^2 + n$$ is an odd integer. It is seen that truth of $$P(n)\Rightarrow$$ the truth of P(n + 1). Therefore, P(n) is true for all

For natural number n, $$2^{n}\, (n - 1) ! <n^{n}$$, if.

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