# Arithmetic and Geometric Sequences

Need some practice with recursion and sequences first? Introduction to Recursion and Sequences

Note:
When you talk about an *arithmetic sequence*,
the word *arithmetic* (in this context)
is pronounced air-ith-ME-tic;
that is, the accent is on the third syllable.

*arithmetic sequence*is a sequence of the form $\,u_n = u_{n-1} + d\,.$

Here, $\,d\,$ is called the *common difference*.

In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant.

## Example

The sequence $\,4\,,$ $\,7\,,$ $\,10\,,$ $\,13\,,$ $\,\ldots\,$ is an arithmetic sequence. The common difference is $\,3\,.$ To go from term to term, you keep adding $\,3\,.$

## Connection Between Arithmetic Sequences and Linear Functions

Recall that *linear functions*
graph as lines, and have a very special property:
equal changes in the input give
rise to equal changes in the output.

Arithmetic sequences have this same
special property:
equal changes in the input
(e.g., moving from term to term)
give rise to equal changes
in the output (determined by the common difference).
Thus, arithmetic sequences
always graph as *points along a line*.

The graph of the sequence $\,4\,,$ $\,7\,,$ $\,10\,,$ $\,13\,,$ $\,\ldots\,$ is the set of points (‘dots’) shown below. When the input is $\,1\,$ (for the first term in the sequence), the output is $\,4\,.$ When the input is $\,2\,$ (for the second term in the sequence), the output is $\,7\,.$ When the input is $\,3\,$ (for the third term in the sequence), the output is $\,10\,,$ and so on.

## Example (arithmetic sequence)

The sequence $\,10\,,$ $\,8\,,$ $\,6\,,$ $\,4\,,$ $\,\ldots\,$ is an arithmetic sequence. The common difference is $\,-2\,.$ To go from term to term, you keep adding $\,-2\,$ (i.e., subtracting $\,2\,$).

The graph of the sequence $\,10\,,$ $\,8\,,$ $\,6\,,$ $\,4\,,$ $\,\ldots\,$ is shown below:

A *geometric sequence*
is a sequence of the form
$\,u_n = r\cdot u_{n-1}\,.$

Here, $\,r\,$ is called the
*common ratio*.

In a geometric sequence, each term is equal to the previous term, multiplied (or divided) by a constant.

## Example (geometric sequence)

The sequence $\,3\,,$ $\,6\,,$ $\,12\,,$ $\,24\,,$ $\,\ldots\,$ is a geometric sequence. The common ratio is $\,2\,.$ To go from term to term, you keep multiplying by $\,2\,.$

## Connection Between Geometric Sequences and Exponential Functions

There is a class of functions,
called *exponential functions*,
that have a very special property:
equal changes in the input cause
the output to be successively multiplied
by a constant.

Geometric sequences have
this same special property:
equal changes in the input
(e.g., moving from term to term)
cause the output to be successively
multiplied by a constant
(determined by the common ratio).
Thus, geometric sequences
always graph as *
points along the graph
of an exponential function.
*

The graph of the sequence $\,3\,,$ $\,6\,,$ $\,12\,,$ $\,24\,,$ $\,\ldots\,$ is shown below:

## Example (geometric sequence)

The sequence $\,100\,,$ $\,50\,,$ $\,25\,,$ $\,12.5\,,$ $\,\ldots\,$ is a geometric sequence. The common ratio is $\,\frac{1}{2}\,.$ To go from term to term, you keep multiplying by $\,\frac{1}{2}\,$ (i.e., dividing by $\,2\,$).

The graph of the sequence $\,100\,,$ $\,50\,,$ $\,25\,,$ $\,12.5\,,$ $\,\ldots\,$ is shown below: