Simplify the following:

$\dfrac{5.(25)^{n + 1} - 25.(5)^{2n}}{5.(5)^{2n + 3} - (25)^{n + 1}}$

Given,

$\Rightarrow \dfrac{5.(25)^{n + 1} - 25.(5)^{2n}}{5.(5)^{2n + 3} - (25)^{n + 1}} \\[1em] = \dfrac{5.(5^2)^{n + 1} - (5^2)(5)^{2n}}{5.(5)^{2n + 3} - (5^2)^{n + 1}} \\[1em] = \dfrac{5.5^{2n + 2} - 5^{2 + 2n}}{5^{2n + 4} - 5^{2n + 2}} \\[1em] = \dfrac{5^{2n + 3} - 5^{2n + 2}}{5^{2n + 4} - 5^{2n + 2}} \\[1em] = \dfrac{5^{2n}.5^3 - 5^{2n}.5^2}{5^{2n}.5^4 - 5^{2n}.5^2} \\[1em] = \dfrac{5^{2n}(5^3 - 5^2)}{5^{2n}(5^4 - 5^2)} \\[1em] = \dfrac{125 - 25}{625 - 25} \\[1em] = \dfrac{100}{600} = \dfrac{1}{6}.$

Hence, $\dfrac{5.(25)^{n + 1} - 25.(5)^{2n}}{5.(5)^{2n + 3} - (25)^{n + 1}} = \dfrac{1}{6}$.