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Formulas and properties of geometric sequence.

Definition
Geometric sequence (geometric progression) — a sequence of numbers b1, b2, b3, ..., in which each member, starting with the second, equal to the product of the previous member and a constant number q (common ratio), where b1 ≠ 0, q ≠ 0.

n -th term of the geometrical sequence is given by

bn = b1 · qn - 1

bn = bn - 1 · q

Common ratio

q bn
bn - 1
A geometric series is the sum of the numbers in a geometric sequence.

Formulas of geometric series

Sn b1 - bn + 1
1 - q

Sn = b1 ·  1 - qn
1 - q

Properties of geometric sequence

bn2 = bn + 1 · bn - 1

An infinite geometric series

If |q| < 1 and n → ∞

S b1
1 - q

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