The reader may notice a connection between this explicit formula and the function $f(x) = \frac {12}{x}-5$ whose domain consists of all real nonzero $x$-values.

• $f(1) = \answer [given]{7}$ and $a_1 = \answer [given]{7}$.
• $f(2) = \answer [given]{1}$ and $a_2 = \answer [given]{1}$.

Similarly, if we evaluate the function $f(x)$ at $x=n$ for any positive integer $n$, we will find the result matches $a_n$. Let’s plot the terms in the sequence and the function on the same axes.

The sequence and the function coincide where they are both defined.

A useful first attempt to associate a function of a real variable to a sequence generated by an explicit formula $a_n=f(n)$ is to try replacing $n$ in with $x$ and let $f(x)$ be the function obtained this way.

Sometimes, we have to work a little harder to find a function of a real variable from which our sequence can be modeled.

Note that while the expressions $(-1)^n$ and $\cos (n \pi )$ may look very different, they generate the same values for each integer $n$. The latter expression, $\cos (\pi x)$ , is necessary to use when finding a function of a real variable that coincides with $(-1)^n$.

The last example suggests that while there is a connection between sequences and functions of real variables, care must be taken when finding a function of a real-variable that coincides with the sequence on their common domains. Another important point to consider is that many different functions of a real variable can agree when they are evaluated at the integers, as the following example will show.

Consider $a_n = 2n+1$ and the functions $f(x) = 2x+1$ and $g(x) = \frac {2x+1}{4}(4+\sin (\pi x))$. The graphs of each are shown below.

Let’s see what happens if use a positive integer $n$ as an input for each function.

• Since $f(x) = 2x+1$, $f(n) = 2n+1$.
• To evaluate $g(n)$, note that since $\sin (k \pi ) = \answer [given]{0}$ for all integers $k$. Thus, $g(n) = \frac {2n+1}{4}(4+\sin (n \pi )) = \frac {2n+1}{\cancel {4}}(\cancel {4}) = 2n+1$.

It should not be too surprising that several functions can be used to represent the same sequence because the sequence is defined only for integers, while the functions above are defined for all real $x$-values in their domains.

While there are many examples of sequences, there are two important types of sequences that arise in applications.

Two important types of sequences

Arithmetic sequences

The first type of sequence we examine is an arithmetic sequence.

Here is a representation of a certain arithmetic sequence.

By looking at the plot above, which type of curve can be be used to model an arithmetic sequence? lines parabolas polynomials exponential curves

Since the average growth rate of a line is constant regardless of the size of the interval chosen, arithmetic sequences correspond to lines.

By requiring that the starting index be $n_0=0$, the sequence $\{a_n\}_{n=0}$, the terms in the sequence are given explicitly by the formula $a_n=\answer [given]{4n-5}$, or recursively by the rule $a_0 = \answer [given]{-5}$ and $a_{n+1} = a_n + \answer [given]{4}$. The difference between subsequent elements in this sequence is always four.

In general, the terms in an arithmetic sequence $\{a_n\}_{n=0}$ in which subsequent terms differ by $m$ can be written as $$a_n = m n + a_0.$$ Alternatively, we could describe an arithmetic sequence recursively, by giving a starting value $a_0$, and using the rule that $a_{n} = a_{n-1} + m$. You should check that this general statement holds for our two examples.

Plots of arithmetic sequences

Recall that arithmetic sequences are those where the difference between neighboring elements is constant. Arithmetic sequences are analogues of lines. Consider the earlier example.

By requiring that the lower index of the sequence be $n_0=1$, we can check out a graph of the sequence.

By noting that the explicit formula for the $n$-th term in this sequence is $a_n = 4n-5$, we can try to find a function of a real variable that coincides with this. The simple rule of replacing $n$ with $x$ works here; we can set $f(x) = 4x-5$ and graph both on the same set of axes.

We see that the common difference is actually the slopethe $y$-coordinate of the $y$-intercept of the line.

Here is an arithmetic sequence that decreases as its index increases.

We can graph the terms.

By setting $n=0$, we can find an explicit formula for the $n$-th term in the sequence; indeed $a_n = \answer [given]{-2n+17}$. As such, we can easily find a function of a real variable $f(x)$ that coincides with the sequence on their common domains by replacing $n$ with $x$ to obtain $f(x) = \answer [given]{-2x+17}$.

We can graph both of these on the same set of axes.

Geometric sequences

A second very important family of sequences we consider are geometric sequences.

Here is an example of a geometric sequence.

By setting the initial index $n_0=0$ , the terms of this sequence can be given explicitly by the formula $a_n=\answer [given]{10\cdot 3^{n}}$, or recursively by the rule $a_0 = \answer [given]{10}$ and $a_{n+1} = \answer [given]{3}\cdot a_n$. The ratio between any subsequent elements of our sequence is three.

A geometric sequence can also decrease as it progresses.

Here is an example of a geometric sequence.

By setting the initial index $n_0=0$ , the terms in this sequence are given by the formula $a_n=\answer [given]{\left (\frac {7}{5}\right )\cdot \left (\frac {1}{2}\right )^{n}}$, or the recursive rule $a_0 = \answer [given]{\frac {7}{5}}$ and $a_{n+1} = \answer [given]{\left (\frac {1}{2}\right )}\cdot a_n$. The ratio between subsequent terms of this sequence is always one half.

In general, a geometric sequence $\{a_n\}_{n=0}$ in which the ratio between subsequent terms is $r$ can be written as $$a_n = a_0 \cdot r^{n}.$$ Alternatively, we could describe a geometric sequence recursively, by giving a starting value $a_0$, and using the rule that $a_{n} = r \cdot a_{n-1}$. As usual, you should check that these general rules hold for the specific examples we’ve considered.

Plots of geometric sequences

Let’s look at plots of some of our special sequences.

Consider the previously illustrated example below.

Let’s plot the terms of the sequence.

The formula for the terms in the sequence in this example was $a_n = 10 \cdot 3^n$. We can easily find a function of a real variable $f(x)$ that coincides with the sequence on their common domains by replacing $n$ with $x$ to obtain $f(x) = \answer [given]{10 \cdot 3^x}$ and graph them both on a common set of axes.

Now let’s consider the second example of a geometric sequence from before, which is illustrated below.

In this example, the explicit formula the terms were given by the formula $a_n = \frac {7}{5} \cdot \left (\frac {1}{2}\right )^n$, so we can find a function of a real variable $f(x)$ that coincides with the sequence by setting $f(x) = \frac {7}{5} \cdot \left (\frac {1}{2}\right )^x$. We can plot these on the same set of axes.

Can all geometric sequences be modeled using an exponential? A third example explores this. Consider the sequence modeled below.

$$a_n = \frac {3}{5} \cdot \left ( -\frac {4}{5}\right )^n = \frac {3}{5} \cdot \left ( - 1 \cdot \frac {4}{5}\right )^n =\frac {3}{5} \cdot (-1)^n \left (\frac {4}{5}\right )^n$$

Replacing $(-1)^n$ with $\cos (\pi x)$ as we did in a previous example gives us a function $$f(x) = \frac {3}{5} \left (\frac {4}{5}\right )^x \cos (\pi x)$$ that agrees with the sequence on their common domains.

Geometric sequences play an important role in the coming sections, so it is useful to have a good feel for what plots of these sequences do!