Simplify: \[ \frac{5\cdot25^{n+1}-25\cdot5^{2n}}{5\cdot5^{2n+3}-25^{n+1}} \]
Solution
\[ 25 = 5^2 \]
\[ = \frac{5\cdot(5^2)^{n+1} – 5^2\cdot5^{2n}}{5\cdot5^{2n+3} – (5^2)^{n+1}} \]
\[ = \frac{5^{2n+3} – 5^{2n+2}}{5^{2n+4} – 5^{2n+2}} \]
\[ = \frac{5^{2n+2}(5 – 1)}{5^{2n+2}(25 – 1)} \]
\[ = \frac{4}{24} = \frac{1}{6} \]
Final Answer:
\[ \boxed{\frac{1}{6}} \]