A powder can be compressed by packing it down. Each time it is packed down it loses $\frac{1}{2}$ then $\frac{1}{4}$ then $\frac{1}{8}$ ... etc. of it's total volume.
This powder is placed in to a container of unit volume and packed down. Then the remaining space is filled with fresh powder and packed down again. The packing action acts on the powder multiple times. So, for example, here is the volume of powder after the first few packings:
$(1)(\frac12) = \frac12$
$(1)(\frac12)(\frac34) + (\frac12)(\frac12) = \frac58$
$(1)(\frac12)(\frac34)(\frac78) + (\frac12)(\frac12)(\frac34) + (\frac38)(\frac12) = \frac{45}{64}$
$(1)(\frac12)(\frac34)(\frac78)(\frac{15}{16}) + (\frac12)(\frac12)(\frac34)(\frac78) + (\frac38)(\frac12)(\frac34) + (\frac{19}{64})(\frac12)$
$\vdots$
I would like to know how much powder is in the container after it has been packed n times, in terms of n.