What is a Geometric Sequence?

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In a geometric sequence, each term is equal to the previous term times a constant, non-zero multiplier called the common factor. Geometric sequences can have a fixed number of terms, or they can be infinite. In either case, the terms of a geometric sequence can rapidly become very large, very negative or very close to zero. Compared to arithmetic sequences, the terms change much more quickly, but while infinite arithmetic sequences increase or decrease steadily, geometric sequences can approach zero, depending on the common factor.

TL;DR (Too Long; Didn't Read)

A geometric sequence is an ordered list of numbers in which each term is the product of the previous term and a fixed, non-zero multiplier called the common factor. Each term of a geometric sequence is the geometric mean of the terms preceding and following it. Infinite geometric sequences with a common factor between +1 and −1 approach the limit of zero as terms are added while sequences with a common factor larger than +1 or smaller than −1 go to plus or minus infinity.

How Geometric Sequences Work

A geometric sequence is defined by its starting number ​a​, the common factor ​r​ and the number of terms ​S​. The corresponding general form of a geometric sequence is:

a, ar, ar^2, ar^3, ... , ar^{S-1}

The general formula for term ​n​ of a geometric sequence (i.e., any term within that sequence) is:

a_n = ar^{n-1}

The recursive formula, which defines a term with respect to the previous term, is:

a_n = ra_{n-1}

An example of a geometric sequence with starting number 3, common factor 2 and eight terms is 3, 6, 12, 24, 48, 96, 192, 384. Calculating the last term using the general form listed above, the term is:

a_8 = 3 × 2^{8-1} = 3 × 2^7 = 3 × 128 = 384

Using the general formula for term 4:

a_4 = 3 × 2^{4-1} = 3 × 2^3 = 3 × 8 = 24

If you want to use the recursive formula for term 5, then term 4 = 24, and a5 equals:

a_5= 2 × 24 = 48

Geometric Sequence Properties

Geometric sequences have special properties as far as the geometric mean is concerned. The geometric mean of two numbers is the square root of their product. For example, the geometric mean of 5 and 20 is 10 because the product 5 × 20 = 100 and the square root of 100 is 10.

In geometric sequences, each term is the geometric mean of the term before it and the term after it. For example, in the sequence 3, 6, 12 ... above, 6 is the geometric mean of 3 and 12, 12 is the geometric mean of 6 and 24, and 24 is the geometric mean of 12 and 48.

Other properties of geometric sequences depend on the common factor. If the common factor ​r​ is greater than 1, infinite geometric sequences will approach positive infinity. If ​r​ is between 0 and 1, the sequences will approach zero. If ​r​ is between zero and −1, the sequences will approach zero, but the terms will alternate between positive and negative values. If ​r​ is less than −1, the terms will trend toward both positive and negative infinity as they alternate between positive and negative values.

Geometric sequences and their properties are especially useful in scientific and mathematical models of real world processes. The use of specific sequences can help with the study of populations that grow at a fixed rate over given periods of time or investments that earn interest. The general and recursive formulas make it possible to predict accurate values in the future based on the starting point and the common factor.

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