Let a complex number $$\alpha ,\,\alpha \ne 1$$, be a root of the equation $${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, where p, q are distinct primes. Show that either $$1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha + {\alpha ^2} + .... + {\alpha ^{q - 1}} = 0$$, but not both together.

Use mathematical induction to show that
$${\left( {25} \right)^{n + 1}} - 24n + 5735$$ is divisible by $${\left( {24} \right)^2}$$ for all $$ = n = 1,2,...$$

Let a, b be positive real numbers. If a, $${{A_1},{A_2}}$$, b are in arithmetic progression, a, $${{G_1},{G_2}}$$, b are in geometric progression and a, $${{H_1},{H_2}}$$, b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$$.

A straight line $$L$$ through the origin meets the lines $$x + y = 1$$ and $$x + y = 3$$ at $$P $$ and $$Q$$ respectively. Through $$P$$ and $$Q$$ two straight lines $${L_1}$$ and $${L_2}$$ are drawn, parallel to $$2x - y = 5$$ and $$3x + y = 5$$ respectively. Lines $${L_1}$$ and $${L_2}$$ intersect at $$R$$. Show that the locus of $$R$$, as $$L$$ varies is a straight line.