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We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.

In the previous section, some sequences were generated by explicit formulas. For instance, let’s consider the sequence $\{a_n\}_{n=1}$ whose $n$-th term is given by $a_n = \frac {12}{n}-5$. We can make a plot of the sequence.

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.

Using the convention above, which function corresponds to the sequence given by the explicit formula $a_n = -n/2+3$ for $n=1,2,3,\dots$?

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.

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.

In general, the terms in an arithmetic sequence $\{a_n\}_{n=0}$ in which subsequent terms differ by $m$ can be written as 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.

#### Geometric sequences

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

A geometric sequence can also decrease as it progresses.

In general, a geometric sequence $\{a_n\}_{n=0}$ in which the ratio between subsequent terms is $r$ can be written as 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.

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.

These examples illustrate an important fact. If the common ratio between successive terms is greater than one, the geometric sequence will increase, but if it is between zero and one, the terms will decrease. In each case, when the common ratio between successive elements of a geometric sequence is positive, which type of curves below model geometric sequences? lines parabolas polynomials exponential curves

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

We can find a formula for the $n$-th term. As before, setting the initial index as $n_0=0$, we can write $a_n = \frac {3}{5} \cdot \left ( -\frac {4}{5}\right )^n$ and plot the terms. The sign of this sequence alternates, so there is no exponential function of a real variable that will model this sequence. However, we may note that

Replacing $(-1)^n$ with $\cos (\pi x)$ as we did in a previous example gives us a function 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!

“Obvious” is the most dangerous word in mathematics” -E.T. Bell