Definition of Geometric Sequences

A geometric progression, also known as a geometric sequence, is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio $r$. For example, the sequence $2, 6, 18, 54, \cdots$ is a geometric progression with common ratio $3$. Similarly $10,5,2.5,1.25,\cdots$ is a geometric sequence with common ratio $\displaystyle{\frac{1}{2}}$. 

Thus, the general form of a geometric sequence is: 

$a, ar, ar^2, ar^3, ar^4, \cdots$

The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by 

${ a }_{ n }=a{ r }^{ n-1 }$

Such a geometric sequence also follows the recursive relation: 

$a_n=ra_{n-1}$ 

for every integer $n\ge 1.$

Behavior of Geometric Sequences

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric series may be negative, resulting in an alternating sequence. An alternating sequence will have numbers that switch back and forth between positive and negative signs. For instance: $1,-3,9,-27,81,-243, \cdots$ is a geometric sequence with common ratio $-3$.

The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:

• Positive, the terms will all be the same sign as the initial term
• Negative, the terms will alternate between positive and negative
• Greater than $1$, there will be exponential growth towards positive infinity ($+\infty$)
• $1$, the progression will be a constant sequence
• Between $-1$ and $1$ but not $0$, there will be exponential decay toward $0$
• $-1$, the progression is an alternating sequence (see alternating series)
• Less than $-1$, for the absolute values there is exponential growth toward positive and negative infinity (due to the alternating sign)

Geometric sequences (with common ratio not equal to $-1$, $1$ or $0$) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as $4, 15, 26, 37, 48, \cdots$ (with common difference $11$). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms $a$, $b$, and $c$ will satisfy the following equation:

${b}^{2}=ac$