Simplify :

$\dfrac{5^{n + 3} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n + 1}}$

Given,

$\dfrac{5^{n + 3} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n + 1}}$

Simplifying the expression :

$\Rightarrow \dfrac{5^{n + 3} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n + 1}} \\[1em] \Rightarrow \dfrac{5^{n + 2 + 1} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n + 1}} \\[1em] \Rightarrow \dfrac{5^{n + 1} \times 5^{2} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n} \times 5 ^{1}} \\[1em] \Rightarrow \dfrac{5^{n + 1}(5^2 - 16)}{5^n (12 - 2 \times 5)} \\[1em] \Rightarrow \dfrac{5^{n + 1 - n}(25 - 16)}{(12 - 10)} \\[1em] \Rightarrow \dfrac{5 \times 9}{(2)} \\[1em] \Rightarrow \dfrac{45}{2} = 22.5$

Hence, $\dfrac{5^{n + 3} - 16 \times 5^{n + 1}}{12 \times 5^n - 2 \times 5^{n + 1}} = 22.5$.