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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

prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
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Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin 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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