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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

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?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
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
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 8

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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