13.1 Sequences and their notations  (Page 6/15)

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.

$a_n=2^n-2$

$a_n=-\frac{16}{n+1}$

$a_n=-{\left(-5\right)}^{n-1}$

$a_n=\frac{2^n}{n^3}$

$a_n=\frac{2n+1}{n^3}$

$a_n=1.25\cdot {\left(-4\right)}^{n-1}$

$a_n=-4\cdot {\left(-6\right)}^{n-1}$

$a_n=\frac{n^2}{2n+1}$

$a_n={\left(-10\right)}^n+1$

$a_n=-\left(\frac{4\cdot {(-5)}^{n-1}}{5}\right)$

$a_n=\{\begin{array}{ll}{(-2)}^n-2\hfill & \text{if }n\text{ is even}\hfill \\ {(3)}^{n-1}\hfill & \text{if }n\text{ is odd}\hfill \end{array}$

$a_n=\{\begin{array}{ll}\frac{n^2}{2n+1}\hfill & \text{if }n\le \text{5}\hfill \\ n^2-5\hfill & \text{if }n\text{ >5}\hfill \end{array}$

$a_n=\{\begin{array}{ll}{(2n+1)}^2\hfill & \text{if }n\text{ is divisible by 4}\hfill \\ \frac{2}{n}\hfill & \text{if }n\text{ is not divisible by 4}\hfill \end{array}$

$a_n=\{\begin{array}{ll}-0.6\cdot 5^{n-1}\hfill & \text{if }n\text{ is prime or 1}\hfill \\ 2.5\cdot {(-2)}^{n-1}\hfill & \text{if }n\text{ is composite}\hfill \end{array}$

$a_n=\{\begin{array}{ll}4(n^2-2)\hfill & \text{if }n\le 3\text{ or }n\text{ > 6}\hfill \\ \frac{n^2-2}{4}\hfill & \text{if }3<n\le 6\hfill \end{array}$

$4, 7, 12, 19, 28,\dots$

$-4,2,-10,14,-34,\dots$

$1,1,\frac{4}{3},2,\frac{16}{5},\dots$

$0,\frac{1-e^1}{1+e^2},\frac{1-e^2}{1+e^3},\frac{1-e^3}{1+e^4},\frac{1-e^4}{1+e^5},\dots$

$1,-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},\dots$

$a_1=9,a_n=a_{n-1}+n$

$a_1=3,a_n=\left(-3\right)a_{n-1}$

$a_1=-4,a_n=\frac{a_{n-1}+2n}{a_{n-1}-1}$

$a_1=-1,a_n=\frac{{\left(-3\right)}^{n-1}}{a_{n-1}-2}$

$a_1=-30,a_n=\left(2+a_{n-1}\right){\left(\frac{1}{2}\right)}^n$

$a_1=\frac{1}{24},{\text{ a}}_2=1,a_n=\left(2a_{n-2}\right)\left(3a_{n-1}\right)$

$a_1=-1,{\text{ a}}_2=5,a_n=a_{n-2}\left(3-a_{n-1}\right)$

$a_1=2,{\text{ a}}_2=10,a_n=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}$

$-2.5,-5,-10,-20,-40,\dots$

$-8,-6,-3,1,6,\dots$

$2,4,12,48,240,\dots$

$35,38,41,44,47,\dots$

$15,3,\frac{3}{5},\frac{3}{25},\frac{3}{125},\cdots$

$6!$