6.1: Sequences

A sequence of numbers in a one-to-one correspondence with the natural numbers \(\{1,2,3,4, \ldots\}\) can be defined in several ways. The terms of the sequence may simply be listed:
\[
\{2,4,8,16,32, \dots\}
\]
A general expression for the sequence may be identified:
\[
a_{n}=2^{n}
\]
In this situation, \(n\) is generally understood to be drawn from the ordered set of natural numbers. In addition, a sequence may be defined recursively. That is to say that each successive term will be defined in relation to the preceding term.
For the example that we're using above, a recursive definition would be as follows:
\[
\begin{array}{c}
a_{1}=2 \\
a_{n}=2 * a_{n-1} \\
\mathrm{or} \\
a_{n+1}=2 * a_{n}
\end{array}
\]
A sequence may be thought of as a function or relation in which the domain is restricted to positive whole numbers.

Examples
Find the first four terms of the given sequence and the 10 th term of the sequence.
1) \(\quad a_{n}=n^{2}+3\)
\[
a_{1}=4, a_{2}=7, a_{3}=12, a_{4}=19, a_{10}=103
\]
2) \(a_{1}=5\)
\[
\begin{array}{l}
a_{n}=a_{n-1}+6 \\
a_{1}=5, a_{2}=11, a_{3}=17, a_{4}=23, a_{10}=59
\end{array}
\]

Finding a general or recursive definition for a sequence can be trickier than just writing out the terms. Common things to look for -

Is this an alternating sequence? That is, do the terms bounce back and forth between positive and negative values. If so, then you will need to include \((-1)^{n}\) or \((-1)^{n+1}\) in your general term.
\[
\begin{array}{l}
\text { Example: }\{-1,2,-3,4, \ldots\} \\
\qquad a_{n}=(-1)^{n}(n)
\end{array}
\]
or
\[
a_{1}=-1
\]
\[
a_{n}=(-1)\left(a_{n-1}\right)+(-1)^{n}
\]
Is there a common difference between the terms? If so, then the sequence behaves much like a linear function and will have a form similar to \(y=m x+b,\) where \(m\) is the common difference.
\[
\begin{array}{l}
\text { Example: }\{5,8,11,14, \ldots\} \\
\qquad a_{n}=3 n+2
\end{array}
\]
or
\[
\begin{array}{l}
a_{1}=5 \\
a_{n}=a_{n-1}+3
\end{array}
\]

Is there a common multiplier? If so, then this should be a power function where a particular base is being raised to the power of \(n\).
\[
\begin{array}{l}
\text { Example: }\{3,15,75,375, \ldots\} \\
\qquad \begin{array}{l}
a_{n}=3 * 5^{n-1} \\
\text { or } \\
a_{1}=3 \\
a_{n}=5 * a_{n-1}
\end{array}
\end{array}
\]
Other patterns to look for are perfect squares and perfect cubes.

Exercises 6.1
Find the first four terms of the given sequence and the 10 th term of the sequence.
1) \(\quad a_{n}=3 n+1\)
2) \(\quad a_{n}=4 n-12\)
3) \(\quad a_{n}=-5 n+3\)
4) \(\quad a_{n}=-2 n+7\)
5) \(\quad a_{n}=2 n^{2}\)
6) \(\quad a_{n}=5 n^{2}-1\)
7) \(\quad a_{n}=(-1)^{n}(4 n)\)
8) \(\quad a_{n}=(-1)^{n+1}\left(\frac{1}{n}\right)\)
9) \(\quad a_{n}=\frac{2^{n}}{3^{n-1}}\)
10) \(\quad a_{n}=\frac{5^{n}}{2^{n+1}}\)
11) \(\quad a_{n}=\frac{(-1)^{n-1}}{2 n+5}\)
12) \(\quad a_{n}=\frac{(-1)^{n}}{3 n-2}\)
13) \(\quad a_{1}=-3 \text{and} a_{n}=a_{n-1}+4 \)
14) \(\quad a_{1}=2 \text{and} a_{n}=a_{n-1}+12\)
15) \(\quad a_{1}=7 \text{and} a_{n}=9-a_{n-1}\)
16) \(\quad a_{1}=-5 \text{and} a_{n}=17-a_{n-1}\)
17) \(\quad a_{1}=1 \text{and} a_{n}=n+a_{n-1}\)
18) \(\quad a_{1}=4 \text{and} a_{n}=n-a_{n-1}\)
19) \(\quad a_{1}=\frac{1}{2} \text{and} a_{n}=\frac{(-1)^{n}}{a_{n-1}}\)
20) \(\quad a_{1}=\frac{2}{5} \text{and} a_{n}=\frac{(-1)^{n+1}}{2 a_{n-1}}\)

For each of the given sequences - find a general term \(a_{n},\) and also find a recursive definition for the sequence.
21) \(\quad\{6,7,8,9,10, \dots\}\)
22) \(\quad\{9,11,13,15,17, \dots\}\)
23) \(\quad\{1,4,7,10,13, \dots\}\)
24) \(\quad\{-5,4,13,22,31, \dots\}\)
25) \(\quad\{-2,6,-18,54, \dots\}\)
26) \(\quad\{5,-10,20,-40,80, \dots\}\)
27) \(\quad\{1,-1,-3,-5,-7, \dots\}\)
28) \(\quad\{-8,-15,-22,-29, \dots\}\)
29) \(\left\{\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \frac{5}{16}, \ldots\right\}\)
30) \(\left\{\frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}, \ldots\right\}\)
31) \(\quad\left\{-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots\right\}\)
32) \(\left\{\frac{1}{2},-\frac{2}{5}, \frac{3}{8},-\frac{4}{11}, \ldots\right\}\)
33) \(\quad\{5,-25,125,-625, \dots\}\)
34) \(\quad\left\{1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \dots\right\}\)
35) \(\quad\left\{1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, \ldots\right\}\)
36) \(\quad\left\{\frac{2}{3}, \frac{9}{4}, \frac{8}{27}, \frac{81}{16}, \frac{32}{243} \dots\right\}\)