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Section exercises

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.

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Describe three ways that a sequence can be defined.

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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.

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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?

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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 13 12 11 10 9 8 7 6 5 4 3 2 1 .

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Algebraic

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

a n = 16 n + 1

First four terms: 8 ,   16 3 ,   4 ,   16 5

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a n = ( 5 ) n 1

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a n = 2 n n 3

First four terms: 2 ,   1 2 ,   8 27 ,   1 4 .

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a n = 1.25 ( 4 ) n 1

First four terms: 1.25 ,   5 ,   20 ,   80 .

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a n = 4 ( 6 ) n 1

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a n = n 2 2 n + 1

First four terms: 1 3 ,   4 5 ,   9 7 ,   16 9 .

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a n = ( 4 ( 5 ) n 1 5 )

First four terms: 4 5 ,   4 ,   20 ,   100

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

a n = { ( 2 ) n 2 if  n  is even ( 3 ) n 1 if  n  is odd

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a n = { n 2 2 n + 1 if  n   5 n 2 5 if  n  >5

1 3 ,   4 5 ,   9 7 ,   16 9 ,   25 11 ,   31 ,   44 ,   59

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a n = { ( 2 n + 1 ) 2 if  n  is divisible by 4 2 n if  n  is not divisible by 4

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a n = { 0.6 5 n 1 if  n  is prime or 1 2.5 ( 2 ) n 1 if  n  is composite

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

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a n = { 4 ( n 2 2 ) if  n   3  or  n  > 6 n 2 2 4 if  3 < n 6

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

4 ,   7 ,   12 ,   19 ,   28 ,

a n = n 2 + 3

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4 , 2 , 10 , 14 , 34 ,

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1 , 1 , 4 3 , 2 , 16 5 ,

a n = 2 n 2 n  or  2 n 1 n

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0 , 1 e 1 1 + e 2 , 1 e 2 1 + e 3 , 1 e 3 1 + e 4 , 1 e 4 1 + e 5 ,

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1 , 1 2 , 1 4 , 1 8 , 1 16 ,

a n = ( 1 2 ) n 1

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

a 1 = 9 ,   a n = a n 1 + n

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a 1 = 3 ,   a n = ( 3 ) a n 1

First five terms: 3 ,   9 ,   27 ,   81 ,   243

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a 1 = 4 ,   a n = a n 1 + 2 n a n 1 1

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a 1 = 1 ,   a n = ( 3 ) n 1 a n 1 2

First five terms: 1 ,   1 ,   9 ,   27 11 ,   891 5

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a 1 = 30 ,   a n = ( 2 + a n 1 ) ( 1 2 ) n

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

a 1 = 1 24 ,  a 2 = 1 ,   a n = ( 2 a n 2 ) ( 3 a n 1 )

1 24 ,  1,  1 4 ,   3 2 ,   9 4 ,   81 4 ,   2187 8 ,   531 , 441 16

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a 1 = 1 ,  a 2 = 5 ,   a n = a n 2 ( 3 a n 1 )

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a 1 = 2 ,  a 2 = 10 ,   a n = 2 ( a n 1 + 2 ) a n 2

2 ,   10 ,   12 ,   14 5 ,   4 5 ,   2 ,   10 ,   12

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

2.5 , 5 , 10 , 20 , 40 ,

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8 , 6 , 3 , 1 , 6 ,

a 1 = 8 , a n = a n 1 + n

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2 ,   4 ,   12 ,   48 ,   240 ,  

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35 ,   38 ,   41 ,   44 ,   47 ,  

a 1 = 35 , a n = a n 1 + 3

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15 , 3 , 3 5 , 3 25 , 3 125 ,

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For the following exercises, evaluate the factorial.

Questions & Answers

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
Practice Key Terms 8

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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