Creating evenly spaced numbers on a log scale (a geometric progression) can easily be done for a given base and number of elements if the starting and final values of the sequence are known, e.g., with numpy.logspace and numpy.geomspace. Now assume I want to define the geometric progression the other way around, i.e., based on the properties of the resulting geometric series. If I know the sum of the series as well as the first and last element of the progression, can I compute the quotient and number of elements?
For instance, assume the first and last elements of the progression are and
and the sum of the series should be equal to
. I know from trial and error that it works out for
n=9 and r≈1.404, but how could these values be computed?
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Answer
You have to solve the following two equations for r and n:
a:= An / Ao = r^(n - 1)
and
s:= Sn / Ao = (r^n - 1) / (r - 1)
You can eliminate n by
s = (r a - 1) / (r - 1)
and solve for r. Then n follows by log(a) / log(r) + 1.
In your case, from s = 50 and a = 15, we obtain r = 7/5 = 1.4 and n = 9.048...
It makes sense to round n to 9, but then r^8 = 15 (r ~ 1.40285) and r = 1.4 are not quite compatible.