Geometric Series

Pre-Calculus · Axiom Academy

LESSON Finite Geometric Series One short formula adds up a geometric sequence of any length — and a single algebraic trick is the reason it works. 1. A Sum That Multiplies by a Constant Ratio In a geometric series, you don't add a fixed amount to get the next term — you multiply by a fixed ratio r . Start at a_1 and the terms march out as A finite geometric series just adds up the first n of them. Each term is the previous one times the ratio r The example above is 3 + 6 + 12 + 24 + 48 , with a_1 = 3 and r = 2 . Each bar is double the last, and the running total climbs to 93 . 2. Why the Formula Works: the S - rS Trick Write the sum as S . Now multiply every term by r to get rS — this just shifts each term one slot to the right. Line the two rows up and subtract: almost everything cancels. all the middle terms cancel — only the ends survive Factor each side and divide by (1-r) to solve for S_n . That one cancellation is the entire reason the formula exists. The original series, a_1 through . Every term , so it lines up one slot over: a_1r through . Each matched middle pair is identical, so it vanishes. Only the first term a_1 and the last shifted term . Solving S_n(1-r) = a_1(1-r^n) gives the result we were after — valid for any number of terms, as long as : Watch it on a shrinking series, with a_1 = 8 and . Each new term is half the last, so the partial sums S_n rise by smaller and smaller steps — creeping toward 16 without ever passing it.

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