terms don't want to stop...

$T_{n} = \dfrac{3^{n}.5^{n}}{5.5^{2n} + 3.3^{2n} - 8.3^{n}5^{n}}$

$\dfrac{1}{5.\frac{5^{n}}{3^{n}} + 3.\frac{3^{n}} {5^{n}} - 8}$

$\dfrac{5^{n}}{3^{n}} = x$

$=\dfrac{1}{5x - 5 + \dfrac{3}{x} - 3}$

$\dfrac{1}{(x-1)(5 - \dfrac{3}{x})}$

$\dfrac{1}{x.(1 - \dfrac{1}{x})(2 + 3(1 - \dfrac{1}{x})}$

$\dfrac{3}{2x}(\dfrac{2}{3(1 - \dfrac{1}{x})(2 + 3(1 - \dfrac{1}{x})})$

$\dfrac{3}{2x}(\dfrac{1}{3(1 - \dfrac{1}{x})} - \dfrac{1}{2 + 3(1 - \dfrac{1}{x})})$

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

$= \dfrac{3}{4}$