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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

the diagram of the digestive system
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Ogenrwot
How does twins formed
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They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
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Genetics is the study of heredity
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the study of living organisms and their interactions with one another and their environments
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the study of living organisms and their interactions with one another and their environment.
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what is the blood cells
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list any five characteristics of the blood cells
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advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
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cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
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is like gone fail us
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cells is the basic structure and functions of all living things
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What is classification
ISCONT Reply
is organisms that are similar into groups called tara
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in what situation (s) would be the use of a scanning electron microscope be ideal and why?
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A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
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Diversity of living thing
ISCONT
what is cell division
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Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
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life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
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Practice Key Terms 8

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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