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Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?
A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.
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.
Yes, both sets go on indefinitely, so they are both infinite sequences.
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.
A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out $\text{13}\cdot \text{12}\cdot \text{11}\cdot \text{10}\cdot \text{9}\cdot \text{8}\cdot \text{7}\cdot \text{6}\cdot \text{5}\cdot \text{4}\cdot \text{3}\cdot \text{2}\cdot \text{1}\text{.}$
For the following exercises, write the first four terms of the sequence.
${a}_{n}={2}^{n}-2$
${a}_{n}=-\frac{16}{n+1}$
First four terms: $-8,\text{}-\frac{16}{3},\text{}-4,\text{}-\frac{16}{5}$
${a}_{n}=-{\left(-5\right)}^{n-1}$
${a}_{n}=\frac{{2}^{n}}{{n}^{3}}$
First four terms: $2,\text{}\frac{1}{2},\text{}\frac{8}{27},\text{}\frac{1}{4}$ .
${a}_{n}=\frac{2n+1}{{n}^{3}}$
${a}_{n}=1.25\cdot {\left(-4\right)}^{n-1}$
First four terms: $1.25,\text{}-5,\text{}20,\text{}-80$ .
${a}_{n}=-4\cdot {\left(-6\right)}^{n-1}$
${a}_{n}=\frac{{n}^{2}}{2n+1}$
First four terms: $\frac{1}{3},\text{}\frac{4}{5},\text{}\frac{9}{7},\text{}\frac{16}{9}$ .
${a}_{n}={\left(-10\right)}^{n}+1$
${a}_{n}=-\left(\frac{4\cdot {(-5)}^{n-1}}{5}\right)$
First four terms: $-\frac{4}{5},\text{}4,\text{}-20,\text{}100$
For the following exercises, write the first eight terms of the piecewise sequence.
${a}_{n}=\{\begin{array}{ll}{(-2)}^{n}-2\hfill & \text{if}n\text{iseven}\hfill \\ {(3)}^{n-1}\hfill & \text{if}n\text{isodd}\hfill \end{array}$
${a}_{n}=\{\begin{array}{ll}\frac{{n}^{2}}{2n+1}\hfill & \text{if}n\text{}\le \text{5}\hfill \\ {n}^{2}-5\hfill & \text{if}n\text{5}\hfill \end{array}$
$\frac{1}{3},\text{}\frac{4}{5},\text{}\frac{9}{7},\text{}\frac{16}{9},\text{}\frac{25}{11},\text{}31,\text{}44,\text{}59$
${a}_{n}=\{\begin{array}{ll}{(2n+1)}^{2}\hfill & \text{if}n\text{isdivisibleby4}\hfill \\ \frac{2}{n}\hfill & \text{if}n\text{isnotdivisibleby4}\hfill \end{array}$
${a}_{n}=\{\begin{array}{ll}-0.6\cdot {5}^{n-1}\hfill & \text{if}n\text{isprimeor1}\hfill \\ 2.5\cdot {(-2)}^{n-1}\hfill & \text{if}n\text{iscomposite}\hfill \end{array}$
$-0.6,-3,-15,-20,-375,-80,-9375,-320$
${a}_{n}=\{\begin{array}{ll}4({n}^{2}-2)\hfill & \text{if}n\text{}\le 3\text{or}n\text{6}\hfill \\ \frac{{n}^{2}-2}{4}\hfill & \text{if}3n\le 6\hfill \end{array}$
For the following exercises, write an explicit formula for each sequence.
$-4,2,-10,14,-34,\dots $
$1,1,\frac{4}{3},2,\frac{16}{5},\dots $
${a}_{n}=\frac{{2}^{n}}{2n}\text{or}\frac{{2}^{n-1}}{n}$
$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}},\dots $
$1,-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},\dots $
${a}_{n}={\left(-\frac{1}{2}\right)}^{n-1}$
For the following exercises, write the first five terms of the sequence.
${a}_{1}=9,\text{}{a}_{n}={a}_{n-1}+n$
${a}_{1}=3,\text{}{a}_{n}=\left(-3\right){a}_{n-1}$
First five terms: $3,\text{}-9,\text{}27,\text{}-81,\text{}243$
${a}_{1}=-4,\text{}{a}_{n}=\frac{{a}_{n-1}+2n}{{a}_{n-1}-1}$
${a}_{1}=-1,\text{}{a}_{n}=\frac{{\left(-3\right)}^{n-1}}{{a}_{n-1}-2}$
First five terms: $-1,\text{}1,\text{}-9,\text{}\frac{27}{11},\text{}\frac{891}{5}$
${a}_{1}=-30,\text{}{a}_{n}=\left(2+{a}_{n-1}\right){\left(\frac{1}{2}\right)}^{n}$
For the following exercises, write the first eight terms of the sequence.
${a}_{1}=\frac{1}{24},{\text{a}}_{2}=1,\text{}{a}_{n}=\left(2{a}_{n-2}\right)\left(3{a}_{n-1}\right)$
$\frac{1}{24},\text{1,}\frac{1}{4},\text{}\frac{3}{2},\text{}\frac{9}{4},\text{}\frac{81}{4},\text{}\frac{2187}{8},\text{}\frac{531,441}{16}$
${a}_{1}=-1,{\text{a}}_{2}=5,\text{}{a}_{n}={a}_{n-2}\left(3-{a}_{n-1}\right)$
${a}_{1}=2,{\text{a}}_{2}=10,\text{}{a}_{n}=\frac{2\left({a}_{n-1}+2\right)}{{a}_{n-2}}$
$2,\text{}10,\text{}12,\text{}\frac{14}{5},\text{}\frac{4}{5},\text{}2,\text{}10,\text{}12$
For the following exercises, write a recursive formula for each sequence.
$-2.5,-5,-10,-20,-40,\dots $
$2,\text{}4,\text{}12,\text{}48,\text{}240,\text{}\dots $
$35,\text{}38,\text{}41,\text{}44,\text{}47,\text{}\dots $
${a}_{1}=35,{a}_{n}={a}_{n-1}+3$
$15,3,\frac{3}{5},\frac{3}{25},\frac{3}{125},\cdots $
For the following exercises, evaluate the factorial.
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