Formulas and properties of geometric sequence

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

b_n = b₁ · q^{n - 1}

b_n = b_{n - 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

b_n² = b_{n + 1} · b_{n - 1}

An infinite geometric series

If |q| < 1 and n → ∞

S =	b1
	1 - q