How can I calculate these using a program or WolframAlpha?

Question

I have this expression: $ S_{n-1} = \frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)} r^{n-1} $

I have this sequence:

1. $ S_{n} $
2. $ \frac{1}{2}(S_{n}) + S_{n-1} $
3. $ \frac{1}{2^2}(S_{n}) + \frac{1}{2}(S_{n-1}) + S_{n-2} $
4. $ \frac{1}{2^3}(S_{n}) + \frac{1}{2^2}(S_{n-1}) + \frac{1}{2}(S_{n-2}) + S_{n-3} $
5. ... stop when $S_{3}$ is reached.

I would like to get the values for each of these (in simplified form with $\pi$ in the expression, not the decimal values), starting with my choice of n.

Answer 1

In Mathematica, One possible solution, probably not the most elegant one, is this: