The value of $$\frac{{3 \times {9^{n + 1}} + 9 \times {3^{2n - 1}}}}{{9 \times {3^{2n}} - 6 \times {9^{n - 1}}}}$$    is equal to-

Solution (By Examveda Team)

$$\eqalign{ & {\text{Expression}} \cr & \frac{{3 \times {9^{n + 1}} + 9 \times {3^{2n - 1}}}}{{9 \times {3^{2n}} - 6 \times {9^{n - 1}}}} \cr & = \frac{{3 \times {{\left( {{3^2}} \right)}^{n + 1}} + {3^2} \times {3^{2n - 1}}}}{{{3^2} \times {3^{2n}} - 6 \times {{\left( {{3^2}} \right)}^{n - 1}}}} \cr & = \frac{{{3^{2n + 2 + 1}} + {3^{2n - 1 + 2}}}}{{{3^{2n + 2}} - 6 \times {3^{2n - 2}}}} \cr & = \frac{{{3^{2n + 3}} + {3^{2n + 1}}}}{{{3^{2n + 2}} - 6 \times {3^{2n - 2}}}} \cr & = \frac{{{3^{2n + 1}}\left( {{3^2} + 1} \right)}}{{{3^{2n - 2}}\left( {{3^4} - 6} \right)}} \cr & = {3^{2n + 1 - 2n + 2}}\left( {\frac{{10}}{{75}}} \right) \cr & = \frac{{{3^3} \times 10}}{{75}} \cr & = \frac{{27 \times 10}}{{75}} \cr & = \frac{{18}}{5} \cr & = 3\frac{3}{5} \cr} $$