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Section 1 Sample Textbook Section

An ordered list of numbers

\begin{equation*} s_1,s_2,s_3,\ldots \end{equation*}

is called a sequence. An individual number in the list is called an entry or term in the sequence.

A linear sequence is a sequence that has the form

\begin{equation*} a,\; a+d,\; a+d+d,\; a+d+d+d, \ldots \end{equation*}

for some constants \(a\) and \(d\text{,}\) so that each entry is equal to \(d\) plus the previous entry. A linear sequence can be written in the simplified form

\begin{equation*} a,\; a+d,\; a+2d,\; a+3d,\ldots . \end{equation*}

For example, the positive odd numbers

\begin{equation*} 1,3,5,7,\ldots \end{equation*}

is a linear sequence with \(a=1\) and \(d=2\text{.}\)

An exponential sequence is a sequence that has the form

\begin{equation*} a, ar, arr, arrr, \ldots \end{equation*}

for some constants \(a\) and \(r\text{,}\) so that each entry is equal to \(r\) times the previous entry. An exponential sequence can be written in the simplified form

\begin{equation*} a,ar,ar^2,ar^3,\ldots . \end{equation*}

For example, the doubling sequence

\begin{equation*} 1,2,4,8,16,\ldots \end{equation*}

is an exponential sequence with \(a=1\) and \(r=2\text{.}\)

Exercises Exercises

1.

The first few terms of each of the following sequences fits a linear pattern, an exponential pattern, or neither. Say which, and explain.

  1. \(\displaystyle 2,6,10,\ldots\)

  2. \(\displaystyle 2,6,18,\ldots\)

  3. \(\displaystyle 2,6,18,20,\ldots\)

Answer.

  1. linear

  2. exponential

  3. neither

2.

Find the 10th terms in the linear and exponential sequences below.

  1. \(\displaystyle 2,\frac{2}{3},\frac{4}{9},\ldots\)

  2. \(\displaystyle 2,-1,-4,\ldots\)

Answer.

  1. \(\displaystyle 2/3^9 \approx 1.02 \times 10^{-4} = 0.000102\)

  2. \(\displaystyle -25\)