What Is a Geometric Sequence?

Definition and Characteristics

A geometric sequence is defined by an initial term a1 and a constant ratio r, where each term equals the previous term times r:
an = a1 * r^(n-1) for n >= 1.
The ratio r drives the behavior: if r > 1, the sequence grows; if 0 < r < 1, it decays; if r < 0, terms alternate in sign. For r = 1, the sequence is constant (an = a1). In real contexts, r often models a growth or decay rate (e.g., finance).

Formulas and Calculations

• n-th term: an = a1 * r^(n-1)
• Sum of first n terms (for r != 1): Sn = a1 * (1 - r^n) / (1 - r)