### Geometric Sequences / Progressions

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. For example:

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:

Now, continue multiplying each product by the common ratio (3 in my example) and writing the result down… over, and over, and over:

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

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 represents the value of the first term in the sequence (5 in the example above), and represents the value of the fifth term in the sequence (405 in the example above).

Continue reading Geometric Sequences and Geometric Series