# Geometric sequence

A geometric sequence is a type of sequence in which each subsequent term after the first term is determined by multiplying the previous term by a constant (not 1), which is referred to as the common ratio. The following is a geometric sequence in which each subsequent term is multiplied by 2:

3, 6, 12, 24, 48, 96, ...

The general form of a geometric sequence can be written as,

a, ar, ar^{2}, ar^{3}, ar^{4},...

where r cannot be equal to 1, and the first term of the sequence, a, scales the sequence. If r is equal to 1, the sequence is a constant sequence, not a geometric sequence.

To determine the n^{th} term of the sequence, the following formula can be used:

a_{n} = ar^{n-1}

where a_{n} is the n^{th} term in the sequence, r is the common ratio, and a is the value of the first term.

Example

Find the 12^{th} term of the geometric series: 1, 3, 9, 27, 81, ...

a_{n} = ar^{n-1} = 1(3^{(12 - 1)}) = 3^{11} = 177,147

Depending on the value of r, the behavior of a geometric sequence varies. If r is not -1, 1, or 0, the sequence will exhibit exponential growth or decay. If r is negative, the sign of the terms in the sequence will alternate between positive and negative.

- r > 1: sequence approaches positive infinity if a > 0 or negative infinity if a < 0
- -1 < r < 1, r ≠ 0: sequence decays exponentially towards 0
- r < -1: sequence grows exponentially approaching infinity (no sign because the sign alternates)

## Geometric sequence vs geometric series

A geometric series is the sum of a finite portion of a geometric sequence. For example, 1 + 3 + 9 + 27 + 81 = 121 is the sum of the first 5 terms of the geometric sequence {1, 3, 9, 27, 81, ...}.

To find the sum of a finite geometric sequence, use the following formula:

where a is the first term in the sequence, r is the common ratio between the terms, and n is the number of terms in the sequence.

Example

Find the sum of the first 12 terms in the geometric series: 1, 3, 9, 27, 81,...