[tex]\it \dfrac{2^{n+3}+2^{n+2}+2^{n+1}+2^n}{3^{n+2}+2\cdot3^{n+1}+3^n}=\dfrac{2^n(8+4+2+1)}{3^n(9+6+1)}=\dfrac{2^n\cdot15}{3^n\cdot16}=\dfrac{2^{n-1}\cdot2\cdot3\cdot5}{3^{n-1}\cdot3\cdot2\cdot8}=\\ \\ \\ =\dfrac{\ \ 5\cdot2^{n-1}\cdot6^{(6}}{8\cdot3^{n-1}\cdot6}=\dfrac{5\cdot2^{n-1}}{8\cdot3^{n-1}},\ \ \forall n\in\mathbb{N}^*[/tex]
