Geometric Series Formula

Algebra 1 · Axiom Academy

LESSON Geometric Series Formula Add up a list that multiplies — and meet the one algebra trick that collapses the whole sum into a single fraction. 1. What Is a Geometric Series? A geometric sequence is a list of numbers where each term is the previous term times a fixed number called the common ratio r . A geometric series is what you get when you add those terms up. Each term is the one before it, so a = 2 and r = 3 . Watch the sequence being built — every new term is born by multiplying by r , and the running total grows as we add each one in. Here is the trick. Write the sum S_n , then write underneath it, shifted over by one slot . Lining them up that way makes almost every term match a partner directly above it. Now subtract the bottom row from the top row. Watch the matched middle terms cancel in pairs — only the very first term and the very last term survive. After everything in the middle cancels, what is left is short: Factor each side, then divide by (1-r) to isolate S_n : With the formula in hand, a sum that would take five additions becomes one substitution. You only need three numbers: the first term a , the ratio r , and how many terms n . a = first term r = common ratio n = number of terms Let's evaluate 2 + 6 + 18 + 54 + 162 ( a=2 , r=3 , n=5 ). Follow each substitution as it drops into the formula and simplifies.

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