Geometric Sequence Calculator

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant factor called the common ratio, denoted r. For example, 2, 6, 18, 54, 162 is a geometric sequence with first term a1 = 2 and common ratio r = 3.

The nth term of a geometric sequence is aₙ = a1 * r^(n-1), where a1 is the first term, r is the common ratio, and n is the position of the term. This formula allows you to find any term directly without calculating all previous terms.

The sum of the first n terms of a geometric sequence depends on the common ratio r. If r = 1, then Sₙ = n * a1. If r ≠ 1, then Sₙ = a1 * (1 - r^n) / (1 - r). This formula is derived from the telescoping property of geometric series.

An infinite geometric series is the sum of all terms in a geometric sequence that continues forever. If the absolute value of the common ratio is less than 1 (|r| < 1), the series converges to a finite sum: S = a1 / (1 - r). If |r| >= 1, the series diverges and has no finite sum.

Geometric sequences model exponential growth and decay. Examples include compound interest (balance multiplied by a constant factor each period), bacterial growth (population multiplying by a factor each generation), radioactive decay (amount divided by a constant factor), and viral spread (number of infected people multiplying each day).