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 larger of \({99^{50}} + {100^{50}}\) and \({101^{50}}\) is ..............

The sum of the coefficients of the plynomial \({\left( {1 + x - 3{x^2}} \right)^{2163}}\) 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 \({C_r} = {{\left( {2n} \right)\,!} \over {r!\left( {2n - r} \right)!}}\,\,\,\,\,r = 0,1,2,\,............,2n\)
P...