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.
In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant.
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
[beautiful math coming... please be patient]
$\,4\,$, $\,7\,$, $\,10\,$, $\,13\,$, $\,\ldots\,$
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.
The graph of the sequence [beautiful math coming... please be patient] $\,10\,$, $\,8\,$, $\,6\,$, $\,4\,$, $\,\ldots\,$ is shown below:
In a geometric sequence, each term is equal to the previous term, multiplied (or divided) by a constant.
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
[beautiful math coming... please be patient]
$\,3\,$, $\,6\,$, $\,12\,$, $\,24\,$, $\,\ldots\,$
is shown below:
The graph of the sequence [beautiful math coming... please be patient] $\,100\,$, $\,50\,$, $\,25\,$, $\,12\,$, $\,\ldots\,$ is shown below: