# 9.1 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.

$6!$

$720$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar