Last year, our teacher made us an exam to check our knowledge in light reflection and refraction. I don't perfectly remember it, but I know that one of the exercises included a ray with its angle of incidence, and let's say 7 media with their respective indexes of refraction. We had to find the last angle of refraction. Even though it's fairly easy to do, it took my classmates a lot of time. Meanwhile I didn't even have time to reach that question: we were very limited on time.
I was thinking if there was another way to find it, a quicker one. Then, while looking at a figure of refraction in my physics book, I came up with an idea: what if we calculate it using only $n_1$, $n_7$, $\Theta_1$ and $\Theta_7$? I'll explain.
Let's say the first medium is air, and I'm calling it $n_1$. The seventh medium is glass, and I'm calling it $n_7$. There are 5 other media between air and glass. The angle of incidence will be $\Theta_1$ and the last angle of refraction, the one inside $n_7$, will be $\Theta_7$. Is it possible to calculate $\Theta_7$ as
$$\Theta_7 = \frac{n_7 \cdot \Theta_1}{n_1},$$
or is it required to calculate each angle of refraction one by one until we reach $\Theta_7$?