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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 gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
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
Please you teach
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Yes
ahmed
Thank you
Dua
give me treganamentry question
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Solve 2cos x + 3sin x = 0.5
shobana 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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