Question 2: \((v)\) \(\quad\) \(\large{\frac{5.( 25)^{n+1} -25.( 5)^{2n}}{5.( 5)^{2n+3} -( 25)^{n+1}} }\)
Solution:
\[\begin{align*}
&= \frac{5 \cdot (25)^{n+1} – 25 \cdot (5)^{2n}}{5 \cdot (5)^{2n+3} – (25)^{n+1}} \\
&= \frac{5 \cdot (5^2)^{n+1} – 5^2 \cdot (5)^{2n}}{(5)^{2n+3+1} – (5^2)^{n+1}} \\
&= \frac{5 \cdot( 5)^{2n+2} – (5)^{2n+2}}{(5)^{2n+4} – (5)^{2n+2}} \\
&= \frac{5^{2n+2} (5-1)}{5^{2n+2}(5^2 -1)} \\
&= \frac{4}{25 -1} \\
&= \frac{4}{24} \\
&= \frac{1}{6}\\
\end{align*}\]