I have a series which I want to have N terms, starting with term A, of which the sum is 1, and have a constant ratio, R between the terms.


0.01283 (A)




Where I wanted R to be the square root of 15

To calculate this, I used

A = 1 / (1 + R(1 + R(1 + R)))

I've labelled it Geometric as that's the closest name I've found, but for the formula, this seems to be the inverse, where the sum is known, but the ratio is not, and the series is finite. What's the proper name for this series, and is there a formula that can calculate the first term (or mth term)?

  • $\begingroup$ Well, the sum $\sum_{i=0}^{N-1}r^i=\frac {1-r^N}{1-r}$ so, if you want the sum to be $1$, you must multiply by the reciprocal of this. Or have I misunderstood the question? $\endgroup$ – lulu Nov 5 '16 at 10:59

You have the sequence $A, AR, AR^2,\dots,AR^{N-1}$, with the constraint $$ A+AR+AR^2+\dots+AR^{N-1}=1. $$ Since the left-hand side can be rewritten (for $R\ne1$) as $$ A\frac{R^N-1}{R-1}, $$ once you fix $N$, you have $$ A=\frac{R-1}{R^N-1} $$

  • $\begingroup$ That's worked. I don't know enough math to know what you did. Do I need to reword the question to make it more searchable? I.E. more accurate names for things $\endgroup$ – Sean Forman Nov 5 '16 at 11:09

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