<s>[INST] Sum of the areas of all n-sided polygons inscribed within each other Imagine you take an $n$-sided regular polygon with side length $x$, and connect the midpoints of each of its sides to construct a smaller but identical $n$-sided regular polygon within it.
Now perform the same process with the smaller polygon, and then with the next one, and the next and so on.
If you continue this, endlessly inscribing smaller and smaller regular $n$-sided polygons, what is the sum of all of their individual areas in terms of $x$ and $n$?
Below is an image describing this question when n = 6:

I know there must be a common ratio between each of the areas, so i'm thinking about turning this into a geometric series of the form $\frac{a}{1-r}$ (since $r$ < 1), where $a$ = the area of the first polygon, however I do not know how to find the specific values for $a$ and $r$ in terms of $x$ and $n$.
Any help with this question would be greatly appreciated!
 [/INST] Let the side length of the outermost polygon and side length of the first constructed polygon be $x_0$ and $x_1$ respectively. The external angles of the polygons will each be $\frac{2\pi}{n}$, so the internal angles will each be $\pi - \frac{2\pi}{n}$. Consider the triangle shown in the diagram below:

Applying the cosine law, we get
$$\begin{align}
(x_1)^2 &= \left( \frac{x_0}{2} \right)^2 + \left( \frac{x_0}{2} \right)^2 - 2\left( \frac{x_0}{2} \right)\left( \frac{x_0}{2} \right) \cos \left( \pi - \frac{2\pi}{n} \right) \\
\\
&= \frac{(x_0)^2}{2} + \frac{(x_0)^2}{2} \cos \left( \frac{2\pi}{n} \right) \\
\\
&= \frac{(x_0)^2}{2} \left[ 1 + \cos \left( \frac{2\pi}{n} \right) \right] \\
\\
&= \frac{(x_0)^2}{2} \left[ 2 \cos^2 \left( \frac{\pi}{n} \right) \right] \\
\\
&= (x_0)^2 \cos^2 \left( \frac{\pi}{n} \right) \\
\\
x_1 &= x_0 \cos\left( \frac{\pi}{n} \right)
\end{align}$$
Thus the common ratio of the sides is $\cos \left( \frac{\pi}{n} \right)$, so the common ratio of the areas will be $\boxed{\cos^2 \left( \frac{\pi}{n} \right)}$
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