The Complex Apartment Complex. A new seven-story apartment building is opening in our town. I was curious when I read that the apartments on the...

The topmost floor has only 1 apartment so the one below has 2 apartments, the one below has 4 apartments, the one below that has 8 etc.

This is a geometric sequence: 1, 2, 4, 8, 16, 32, 64.

One approach to this problem would be to simply add all those numbers:

`1+2+4+8+16+32+64=127`

The other approach (which is especially good for large numbers and long sequences) is to use formula for sum of the first `n` terms of geometric series.

`S=a(1-r^n)/(1-r),`  `r ne 1`

where `a` is the first term and `r` is the common ratio of the series.

In this case `a=1` and `r=2,` hence we have

`S=(1-2^7)/(1-2)=127`

The result is, of course, the same. There are 127 apartments in the building.