Computing Geometric Series

Pre-Calculus · Axiom Academy

EXAMPLE Computing Geometric Series Summing a finite geometric series with the constant-ratio sum formula. Find the sum of the geometric series . The first term is 3 , each term is double the one before, and the last term is 768 . Each bar is double the one before, so the terms rocket from 3 to 768 in just 9 steps. Adding them one by one is slow — the sum formula does it in a single line. Nice work — you summed a series that climbs to 768 without adding all nine terms, by plugging the first term, the ratio, and the count into one formula. Sum formula: — the first term a times one minus the ratio to the n th power, all over one minus the ratio. Count the terms first: rearrange to find n . Here gives , so n = 9 — don't forget the -1 in the exponent. Sigma notation is the same job: has a = 5 , r = 2 , n = 10 , so . It works when the ratio is a fraction too: a ball dropped from 20 m that rebounds to of its height each bounce has bounce heights — a geometric series with . The same formula sums the first six bounces to about 49.32 m (total travel m, counting each bounce up and down). Whenever a list of numbers grows (or shrinks) by a constant ratio , you can replace the long addition with one quick application of .

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