Using mathematical induction prove that for every integer $$n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)$$ is divisible by $${2^{n + 2}}$$ but not by $${2^{n + 3}}$$.

For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$$

The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma $$ lie.

Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $$\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)$$ to the circle $$\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt 2 a)\,x - (1 - \sqrt 2 a)\,y = 0$$.