Geometric Sequences and Geometric Series

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:

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).

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Arithmetic Sequences and Arithmetic Series

Arithmetic Sequences / Progressions

The terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic 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 an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down.  For example:

5

Now pick a second number, any number (I’ll choose 3), which we will call the common difference. Now add the common difference to the first number, then write their sum down to the right of the first number:

5,~8

Now, continue adding the common difference to the sum and writing the result down… over, and over, and over:

5,~8,~11,~14,~17,~20,~23,~26,~29, ...

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:

ArithSequence

Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting 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 of the sequence 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 of the sequence (5 in the example above), and a_7 represents the value of the seventh term of the sequence (23 in the example above).

Continue reading Arithmetic Sequences and Arithmetic Series