The terms “sequence” and “progression” are interchangeable. A “geometric sequence” is the same thing as a “geometric progression”. This post uses the term “sequence”… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create a geometric sequence (also known as a geometric progression).
Pick a number, any number, and write it down. Â For example:
5
Now pick a second number, any number (I’ll choose 3), which we will call the common ratio. Now multiply the first number by the common ratio, then write their product down to the right of the first number:
5,~15
Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:
5,~15,~45,~135,~405,~1,215, ...
By following this process, you have created a “Geometric Sequence”, a sequence of numbers in which the ratio of every two successive terms is the same.
Vocabulary and Notation
In the example above, 5 is the first term (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation
a_1
This notation is read as “A sub one” and means: the 1st value in the sequence or progression represented by “a”. The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence.  So a_1 represents the value of the first term in the sequence (5 in the example above), and a_5 represents the value of the fifth term in the sequence (405 in the example above).
The Common Ratio
Since all of the terms in a Geometric Sequence must be the same multiple of the term that precedes them (3 times the previous term in the example above), this factor is given a formal name (the common ratio) and is often referred to using the variable R (for Ratio). If you multiply any term by this value, you end up with the value of the next term.
For an existing Geometric Sequence, the common ratio can be calculated by dividing any term by its preceding term:
\dfrac{a_2}{ a_1},~~\text{or}~~\dfrac{a_7}{a_6},~~\text{etc.}
Every Geometric Sequence has a common ratio between consecutive terms. Â Examples include:
1,~2,~4,~8,~16,~...
27,~-9,~3~,~-1,~...
1,~0.1,~0.01,~0.001,~...
The common ratio can be positive or negative. It can be a whole number, a fraction, or even an irrational number. No matter what value it has, it will be the ratio of any two consecutive terms in the Geometric Sequence.
Therefore, to test if a sequence of numbers is a Geometric Sequence, calculate the ratio of successive terms in various locations within the sequence. If you calculate the same ratio between any two adjacent terms chosen from the sequence (be sure to put the later term in the numerator, and the earlier term in the denominator), then the sequence is a Geometric Sequence. One of the series shown above can be used to demonstrate this process:
27,~-9,~3~,~-1,~...
\dfrac{-9}{27}=-\dfrac{1}{3}
\dfrac{3}{-9}=-\dfrac{1}{3}
\dfrac{-1}{3}=-\dfrac{1}{3}
Since the ratio between adjacent terms was always equal to the same number (negative one third), this is a Geometric Sequence.
Step-by-step explanation:
