Understanding Geometric Sequences

In a geometric sequence each term is obtained by multiplying the preceding term by a constant, called the common ratio (r). The sequence can be finite or infinite, and the values can grow, shrink, or oscillate depending on r.

TL;DR

A geometric sequence is an ordered list where each term equals the previous term times a non‑zero common ratio. If |r|<1 the terms converge to zero; if |r|>1 they diverge to infinity.

Definition & Formulae

The sequence starts with an initial term a and is expressed as: a, ar, ar2, ar3, …, arS-1. The nth term is given by: an = a·rn-1. A recursive form is an = r·an-1.

Example: a=3, r=2, S=8 → 3, 6, 12, 24, 48, 96, 192, 384. The 8th term is a8 = 3·27 = 384.

Key Properties

• Each interior term is the geometric mean of its neighbors.
• When r>1, an infinite sequence diverges to +∞.
• When 0
• When –1
• When r<–1, the sequence alternates signs and diverges to ±∞.

Real‑World Applications

Geometric sequences model exponential growth or decay, such as:

• Population growth or radioactive decay.
• Compound interest in finance.
• Signal attenuation in engineering.

Accurate forecasting in these domains relies on the general and recursive formulas, enabling predictions from a single known term and the common ratio.

For an in‑depth mathematical treatment, refer to Introductory Mathematical Sequences by J. Smith, 2020.