t_n = \frac{n.2^n}{(n+2)!} = \frac{2^n}{(n+1)!}-\frac{2^{n+1}}{(n+2)!}
and you have a telescopic summation
sum of the series
\frac{1.2}{3!}+\frac{2.2^2}{4!}+\frac{3.2^3}{5!}+............\infty
t_n = \frac{n.2^n}{(n+2)!} = \frac{2^n}{(n+1)!}-\frac{2^{n+1}}{(n+2)!}
and you have a telescopic summation