It'll be obvious when I see the answer, but I'm too tired/thick to figure it out. If I have a certain time period, and I want to divide it into a given number of geometrically increasing (or decreasing) durations so that the ratio of interval(n)/interval(n-1) is constant, then how do I figure out the ratio and/or the starting duration of the first interval?
For example if you wanted to divide a 15-second period into 4 increasing durations you might get the geometric series 1, 2, 4, 8 (where the ratio is 2, and start duration is 1 second)
I think it's something to do with log(T)/N, where T is the total time and N is the number of divisions, but that's as far as I've got.
Say you have an interval of length L and you want to divide it into N intervals $I_k$ s.t. $I_k/I_{k-1} =q$ for $k=1,..N-1$. Then the length of each interval can be written as $cq^k, k=0,...,N-1$, where $c>0$ is some constant and $$ L= \sum_{k=0}^{N-1} cq^k = c \frac{1-q^N}{1-q}$$ You would have to solve this for fixed N, which is not trivial when N>2.