Chapter 11

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

Describe three ways that a sequence can be defined.

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

What happens to the terms \(a_{n}\) of a sequence when there is a negative factor in the formula that is raised to a power that includes \(n ?\) what is the term used to describe this phenomenon?

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

For the following exercises, write the first four terms of the sequence. $$ a_{n}=2^{n}-2 $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\frac{16}{n+1} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-(-5)^{n-1} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{2^{n}}{n^{3}} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{2 n+1}{n^{3}} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=1.25 \cdot(-4)^{n-1} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-4 \cdot(-6)^{n-1} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n^{2}}{2 n+1} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=(-10)^{n}+1 $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\left(\frac{4 \cdot(-5)^{n-1}}{5}\right) $$

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{(-2)^{n}-2} & {\text { if } n \text { is even }} \\ {(3)^{n-1}} & {\text { if } n \text { is odd }}\end{array}\right. $$

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{\frac{n^{2}}{2 n+1}} & {\text { if } n \leq 5} \\\ {n^{2}-5} & {\text { if } n>5}\end{array}\right. $$

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{(2 n+1)^{2}} & {\text { if } n \text { is divisible by } 4} \\ {\frac{2}{n}} & {\text { if } n \text { is not divisible by } 4}\end{array}\right. $$

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{4\left(n^{2}-2\right)} & {\text { if } n \leq 3 \text { or } n>6} \\ {\frac{n^{2}-2}{4}} & {\text { if } 3

For the following exercises, write an explicit formula for each sequence. $$4, 7, 12, 19, 28, …$$

For the following exercises, write an explicit formula for each sequence. $$ 0, \frac{1-e^{1}}{1+e^{2}}, \frac{1-e^{2}}{1+e^{3}}, \frac{1-e^{3}}{1+e^{4}}, \frac{1-e^{4}}{1+e^{5}}, \ldots $$

For the following exercises, write an explicit formula for each sequence. $$ 1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \ldots $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=9, a_{n}=a_{n-1}+n $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=3, a_{n}=(-3) a_{n-1} $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=-4, a_{n}=\frac{a_{n-1}+2 n}{a_{n-1}-1} $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=-1, a_{n}=\frac{(-3)^{n-1}}{a_{n-1}-2} $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=-30, a_{n}=\left(2+a_{n-1}\right)\left(\frac{1}{2}\right)^{n} $$

For the following exercises, write the first eight terms of the sequence. $$ a_{1}=\frac{1}{24}, a_{2}=1, a_{n}=\left(2 a_{n-2}\right)\left(3 a_{n-1}\right) $$

For the following exercises, write the first eight terms of the sequence. $$ a_{1}=-1, \mathrm{a}_{2}=5, a_{n}=a_{n-2}\left(3-a_{n-1}\right) $$

For the following exercises, write the first eight terms of the sequence. $$ a_{1}=2, \mathrm{a}_{2}=10, a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}} $$

For the following exercises, write a recursive formula for each sequence. $$ -2.5,-5,-10,-20,-40, \dots $$

For the following exercises, write a recursive formula for each sequence. $$ -8,-6,-3,1,6, \dots $$

For the following exercises, write a recursive formula for each sequence. $$ 2,4,12,48,240, \dots $$

For the following exercises, write a recursive formula for each sequence. $$ 35,38,41,44,47, \dots $$

For the following exercises, write a recursive formula for each sequence. $$ 15,3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \dots $$

For the following exercises, evaluate the factorial. $$6 !$$

For the following exercises, evaluate the factorial. $$ \left(\frac{12}{6}\right) ! $$

For the following exercises, evaluate the factorial. $$ \frac{12 !}{6 !} $$

For the following exercises, evaluate the factorial. $$ \frac{100 !}{99 !} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n !}{n^{2}} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{3 \cdot n !}{4 \cdot n !} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{100 \cdot n}{n(n-1) !} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\frac{(-1)^{n}}{n}+n $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\left\\{\begin{array}{ll}{\frac{4+n}{2 n}} & {\text { if } n \text { is even }} \\ {3+n} & {\text { if } n \text { is odd }}\end{array}\right. $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{1}=2, a_{n}=\left(-a_{n-1}+1\right)^{2} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\frac{(n+1) !}{(n-1) !} $$

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: \(\bullet\) On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. \(\cdot\) Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1} .\) Press [ENTER]. \(\cdot\) Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. Find the first five tems of the sequence \(a_{1}=\frac{87}{111}, a_{n}=\frac{4}{3} a_{n-1}+\frac{12}{37}\) . Use the \( > \) Frac feature to give fractional results.

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: \(\bullet\) On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. \(\cdot\) Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1} .\) Press [ENTER]. \(\cdot\) Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. Find the \(15^{\text { th }}\) term of the sequence \(a_{1}=625, \quad a_{n}=0.8 a_{n-1}+18\)

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: \(\bullet\) On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. \(\cdot\) Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1} .\) Press [ENTER]. \(\cdot\) Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. Find the first ten terms of the sequence \(a_{1}=8, a_{n}=\frac{\left(a_{n-1}+1\right) !}{a_{n-1} !}\)

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: \(\bullet\) On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. \(\cdot\) Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1} .\) Press [ENTER]. \(\cdot\) Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. Find the tenth term of the sequence \(a_{1}=2, a_{n}=n a_{n-1}\)