Let $y_k=\dfrac{16}{3}y_{k-1}-\dfrac{5}{3}y_{k-2}$
with $y_0=1$ and $y_1=\dfrac{1}{2}$ for $k \geq 2, k \in \mathbb{N}$.
To find the solution $\lbrace y_k \rbrace_{k \in \mathbb{N_0}}$ I used:
Firstly, I computed it for $k=2$:
$y_2=\dfrac{16}{3}\dfrac{1}{2}-\dfrac{5}{3}1=1$
What has to be done next? Induction isn't helpful here.
I don't see how can it be found for $\lbrace y_k \rbrace_{k \in \mathbb{N_0}}$.
$$y_k=\frac{{{5}^{k}}}{28}+\frac{{{3}^{3-k}}}{28}$$
$$[1,1/2,1,9/2,67/3,2009/18,15067/27,452009/162,3390067/243,101702009/1458,762765067/2187]$$
Solution details:
Char. equation is $${{t}^{2}}-\frac{16 t}{3}+\frac{5}{3}=0$$ $$t_1=5,\quad t_2=\frac13$$ Then solution is $$y_k=A\,5^k+B\,\left(\frac13\right)^k$$ From initial conditions $$A\,5^0+B\,\left(\frac13\right)^0=1,\\ A\,5^1+B\,\left(\frac13\right)^1=\frac12$$ $\Rightarrow$ $$A=\frac{1}{28},\quad B=\frac{27}{28}$$ $\Rightarrow$ $$y_k=\frac{{{5}^{k}}}{28}+\frac{{{3}^{3-k}}}{28}$$
see for example https://www.youtube.com/watch?v=aHw7hAAjbD0
