Principle of Ma...

The value of $$\displaystyle \frac{1^2}{1.3} + \frac{2^2}{3 . 5}+\dots+ \frac{n^2}{(2n - 1)(2 n + 1)}$$ is

By mathamatical induction $$n(n^2 - 1)$$ is divisible by

If $$n$$ is an odd positive integer, then $$a^n + b^n$$ is divisible by

If $$a_1 = 1, a_{n+1} = \dfrac{1}{n+1}a_n,\forall n \geq 1$$, then $$a_n \ =$$

$$3 + 13 + 29 + 51 + 79 + ...$$ to $$n$$ terms $$=$$

The value of $$\displaystyle \frac{1}{1.2.3} + \frac{1}{2.3.4}+ ....+ \frac{1}{n (n + 1) (n + 2)}$$ is 

The value of $$\displaystyle \tan^{-1} \left ( \frac{1}{3} \right ) + \tan^{-1} \left ( \frac{1}{7} \right ) +\dots+ \tan^{-1} \left ( \frac{1}{n^2 + n + 1} \right )$$ is

By mathematical induction $$p^{n+1} + (p+1)^{2n -1}$$ is divisible by

Using the principle of mathematical induction, find $$tan  \alpha  + 2  tan   2 \alpha   + 2^2  tan  2^2  \alpha + ....$$ to $$n$$ terms:

If $$p$$ is a prime number, then $$n^p -n$$ is divisible by $$p$$ for all $$n$$, where