# 11.1 Sequences and their notations  (Page 6/15)

 Page 6 / 15

## Verbal

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 $\text{13}\cdot \text{12}\cdot \text{11}\cdot \text{10}\cdot \text{9}\cdot \text{8}\cdot \text{7}\cdot \text{6}\cdot \text{5}\cdot \text{4}\cdot \text{3}\cdot \text{2}\cdot \text{1}\text{.}$

## Algebraic

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:

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

${a}_{n}=\frac{{2}^{n}}{{n}^{3}}$

First four terms: .

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

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

First four terms: .

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

${a}_{n}=\frac{{n}^{2}}{2n+1}$

First four terms: .

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

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

First four terms:

For the following exercises, write the first eight terms of the piecewise sequence.

$-0.6,-3,-15,-20,-375,-80,-9375,-320$

For the following exercises, write an explicit formula for each sequence.

${a}_{n}={n}^{2}+3$

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

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

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

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

${a}_{n}={\left(-\frac{1}{2}\right)}^{n-1}$

For the following exercises, write the first five terms of the sequence.

First five terms:

First five terms:

For the following exercises, write the first eight terms of the sequence.

For the following exercises, write a recursive formula for each sequence.

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

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

${a}_{1}=-8,{a}_{n}={a}_{n-1}+n$

${a}_{1}=35,{a}_{n}={a}_{n-1}+3$

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

For the following exercises, evaluate the factorial.

$6!$

$720$

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this