Geometric Sequence Formula

Algebra 1 · Axiom Academy

LESSON Geometric Sequence Formula Each term is the previous one times the same factor r — so the nth term is the first term with that jump applied exactly (n−1) times. 1. A Geometric Sequence Is a Chain of ×r A geometric sequence is a list of numbers where each term comes from the previous one by multiplying by the same factor . A bacteria culture that doubles every hour reads 1, 2, 4, 8, 16, … — every step is a ×2 . The factor we multiply by each time is the common ratio , written r . To find it, divide any term by the one before it — every quotient comes out the same. For 3, 6, 12, 24, 48 the animation shows each division collapsing to the same value. 3. Counting the Jumps Builds the Formula Start at the first term a_1 and apply ×r one step at a time. Watch the jump counter : reaching a_2 took one ×r, a_3 took two, a_4 took three. The number of jumps is always one less than the term number — and that count is the exponent on r . Putting the jump count into the exponent gives the explicit geometric sequence formula — it jumps straight to any term without listing the ones before it. a_n = nth term · a_1 = first term · r = common ratio · n = which term The animation works a_5 for the sequence 2, 6, 18, 54, …: it highlights exactly 4 of the ×r jumps (because n-1 = 4 ), so , then fills in the numbers. Another: find the 7th term of 5, 10, 20, 40, … Here a_1 = 5 , r = 2 , n = 7 , so the exponent is 7-1 = 6 :

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