Principle of Ma...

Let $$a, b, c$$ and $$d$$ be any four real numbers. Then, $$a^n+b^n=c^n+d^n$$ holds for any natural number $$n$$, if

For any $$+$$ve integer $$n, n^3 + 2n$$ is always divisible by

The last digit in $$\displaystyle { 7 }^{ 300 }$$ is:

For any integer $$n\ge 1$$, the sum $$\displaystyle\sum _{ k=1 }^{ n }{ k\left( k+2 \right)  } $$ is equal to

Let $$A = \begin{pmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix}$$. Then for positive integer $$n, A^{n}$$ is

The integer next above $$(\sqrt{3}+1)^{2n}$$ contains

If $${a_{1,}}{a_{2,}}{a_{3,}}........{a_{2n + 1}}$$ are in A.P. then$$\frac{{{a_{2n + 1}} - {a_1}}}{{{a_{2n + 1}} + {a_1}}} + \frac{{{a_{2n + 1}} - {a_2}}}{{{a_{2n + 1}} + {a_2}}} + ...... + \frac{{{a_{2n + 1}} - {a_n}}}{{{a_{2n + 1}} + {a_n}}}$$ 

$$33!$$ is divisble by $${2^n}$$, then $$n$$=......

Let $$P(n)$$ be the statement $$2^{n}<n!$$ where $$n$$ is a natural number, then $$P(n)$$ is true for:

For all positive integrals $$10^{n}+3^{4n+2}+8$$ is divisible by