Mathematical Induction and Binomial Theorem

Prove that ${7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)$ is divisible by 25 for any natural number $n$.

The sum of the coefficients of the plynomial ${\left( {1 + x - 3{x^2}} \right)^{2163}}$ is ...............

The larger of ${99^{50}} + {100^{50}}$ and ${101^{50}}$ is ..............

Given that ${C_1} + 2{C_2}x + 3{C_3}{x^2} + ......... + 2n{C_{2n}}{x^{2n - 1}} = 2n{\left( {1 + x} \right)^{2n - 1}}$
where ${C_r} = {{\left( {2n} \right)\,!} \over {r!\left( {2n - r} \right)!}}\,\,\,\,\,r = 0,1,2,\,............,2n$
Prove that ${C_1}^2 - 2{C_2}^2 + 3{C_3}^2 - ............ - 2n{C_{2n}}^2 = {\left( { - 1} \right)^n}n{C_n}.$