Answer
The value is approaching $w_1 = 1$.
Therefore, the point $z_1 = 2+i$ should be colored red.
Work Step by Step
$f(z) = \frac{2z^3+1}{3z^2}$
$z_1 = 2+i$
We can evaluate the function for $z_1$:
$z_2 = f(z_1)$
$z_2 = \frac{2(2+i)^3+1}{3(2+i)^2}$
$z_2 = \frac{2(2+11i)+1}{3(3+4i)}$
$z_2 = (\frac{5+22i}{9+12i})(\frac{9-12i}{9-12i})$
$z_2 = \frac{309+138i}{225}$
We can evaluate the function for $z_2$:
$z_3 = f(z_2)$
$z_3 = \frac{2(\frac{309+138i~}{225}~)^3+1}{3(\frac{309+138i}{225})^2}$
$z_3 = \frac{2(\frac{11849841+36901062i}{11390625}~)+1}{3(\frac{76437+85284i}{50625})}$
$z_3 = \frac{(\frac{35090307+73802124i}{11390625})}{(\frac{229311+255852i}{50625})}$
$z_3 = [\frac{(\frac{35090307+73802124i}{225})}{229311+255852i}]~[\frac{229311-255852i}{229311-255852i}]$
$z_3 = \frac{(\frac{26929014418125+7945713630000i}{225})}{118043780625}$
$z_3 = 1.014+0.2992i$
We can see that the value is approaching $w_1 = 1$. Therefore, the point $z_1 = 2+i$ should be colored red.
