Full answer: In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt

In Exercises 1-12. write the first four terms of each sequence whose general term is given. 0 11 - 3n + 2

In Exercises 1-12. write the first four terms of each sequence whose general term is given. 0 11 - 4n - 1

In Exercises 1-12. write the first four terms of each sequence whose general term is given. an - 3"

In Exercises 1-12. write the first four terms of each sequence whose general term is given. - (t)"

In Exercises 1-12. write the first four terms of each sequence whose general term is given. 0 11 - (- 3)"

In Exercises 1-12. write the first four terms of each sequence whose general term is given. - (-D

In Exercises 1-12. write the first four terms of each sequence whose general term is given. a. -(- l}"(n +3)

In Exercises 1-12. write the first four terms of each sequence whose general term is given. a. -(- 1)"+1(n + 4)

In Exercises 1-12. write the first four terms of each sequence whose general term is given. a,. - ;;-+'4

In Exercises 1-12. write the first four terms of each sequence whose general term is given. a, - --n+5

In Exercises 1-12. write the first four terms of each sequence whose general term is given. - 1)"" all - 2" 1

In Exercises 1-12. write the first four terms of each sequence whose general term is given. - 2" + 1

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. a 1 - 7anda, - a, _1 +5 forn 2 2

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. a 1 - 12 and a, - a,_1 + 4 for n a 2

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. n1 - 3 and a" - 4a,_1 for 11 :::: 2

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. a 1 - 2 and"" - 5a, _, for n ::=: 2

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. a 1 - 4anda, - 2a, _1 + 3 forn ::=: 2

The sequences in Exercises 13- 18 are defined usitJg recursio11 Jonmdas. Write the first four terms of each sequenc.e. a 1 - 5 and a,. - 3a,._1 - 1 for 11 a 2

In Exercises 19-22, the general term of a sequence is given and involves a [attorial. Write the first four tenus of each sequence. all - 1

In Exercises 19-22, the general term of a sequence is given and involves a [attorial. Write the first four tenus of each sequence. (11 + 1)! all - ll. IJ2

In Exercises 19-22, the general term of a sequence is given and involves a [attorial. Write the first four tenus of each sequence. n. - 2(11 + 1)!

In Exercises 19-22, the general term of a sequence is given and involves a [attorial. Write the first four tenus of each sequence. n. - - 2(11 - 1)!

In Exercises 23-28. evaluate each factorial expression. .!2.!. 18! 15!

In Exercises 23-28. evaluate each factorial expression. 8! 15! i6i

In Exercises 23-28. evaluate each factorial expression. 2ii'4!

In Exercises 23-28. evaluate each factorial expression. 20! 2ii8!

In Exercises 23-28. evaluate each factorial expression. (11 + 2)! n

In Exercises 23-28. evaluate each factorial expression. 211 + 1)! n! (211)!

In Exercises 29-42, find each indicated sum. 2: 5i i- 1

In Exercises 29-42, find each indicated sum. 2: 7i i- 1

In Exercises 29-42, find each indicated sum. 2: ' 2i1

In Exercises 29-42, find each indicated sum. ' i- 1 2: i' i- 1

In Exercises 29-42, find each indicated sum. 2: k(k + 4)

In Exercises 29-42, find each indicated sum. 2; (k - 3)(k + 2)

In Exercises 29-42, find each indicated sum. J ( 1y J ( 1y 2: -- 2: -::: i- 1 2

In Exercises 29-42, find each indicated sum. 2: -::: i- 1 2 i- 1 .)

In Exercises 29-42, find each indicated sum. 2: 11 i- 5

In Exercises 29-42, find each indicated sum. 2: 12 i- 3

In Exercises 29-42, find each indicated sum. (- 1)' 2; - .- i- 0 t!

In Exercises 29-42, find each indicated sum. J (- 1)'+1 ~(i + 1)!

In Exercises 29-42, find each indicated sum. ' i! (i + 2)! ~ (i 1 )!

In Exercises 29-42, find each indicated sum. (i + 2)

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 12 + 22 + 32 + ... + 15'

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 1' + 24 + 3' + ... + 12'

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 2 + 22 + 23 + .. . + 2 11

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 5 + 52 + 53 + .. . + 5 12

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 1 + 2 + 3 + ... + 30

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 1 +2+3+ .. +40

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 2 3 4 ~ , .. +--14 + 1

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 3 + 4 + 5 + . .. + "i6""+2

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 4+ - ..t. - + + - 2 3 II

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. I 2 3 II 52. - + - + - + +- 99293 9"

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. 1 + 3 + 5 + + (211 - I)

In Exercises 43-54, express each sum using summation notation. Use I as the lower Hmil of summatioiJ and i for the index of summation. a + ar + ar2 + ... + arn- l

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. 5 + 7 + 9 + 11 + ... + 31

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. 6 + 8 + 10 + 12 + +32

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. a + nr + ar 2 + .. + ar t2

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. a + ar + ar2 ..!. ~ir "

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. a + (a + d) + (a ~21!) + + (a + 11d)

In Exercises 55-60. express each sum using summation notation. Use a lower limit of Slmuuation of your choice and k for the index of summation. (a + d) + (a + a2 l111::t. . a+ d" )

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -5 ' (a!+ 1)

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -6 2:(bl - 1)

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -7 2: ' (2a; + b;)

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -8 2: (a;+ 3b1 )

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -9 ~ ' (a)' b;

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -10 ~ ' ("b; J

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -11 2:' ' 1 + 2; b?

In Exercises 61-68. use the graphs oflli,J a11d lb.J to ji11d eadt indicated sum. The Graph of {a,J The Graph of { b,} b, 5 5 4 4 3 3 2 2 1 1 - 1 2 3 4 5 " " - 1 2 3 4 s -2 -2 -3 -3 -4 -4 -5 -12 2: ai - 2: b?

One in one hundred is ~?rati o of individuals in the U.S. population with autisticFdiordcr, better known as autism. 1lte disorder involves ~~ e deficils in language, social bonding, and imaginati6n, and is often accompanied by mental retardation. The f"bar graph at the top of lhe next column shows the number of autism cases diagnosed in the United States from 2001 lhrough 2008. Autism Cases Diagnosed in the United States 300 260 100 Year Souroe: Lilienfeld et al., P:iycho/ogy, Second Edition, Pearson, 201 1. Let On re.present the number of autism cases diagnosed in the United States. in thousands, 11 years afte r 2000. a. Use the numbe rs given in the graph to find and interpret 2;a;. ,_, b. ll1e finite sequence whose general term is an - 2811 + 63, where n - 1, 2, 3, ... , 8, models the number of autism cases diagnosed in lhe United States., in thousands. s 11 years after 2000. Use this model to find 2;a1 Does this i- i underestimate or overestimate the actual sum in part (a)? By how much?

1l1e bar graph shows the average state cigarelle tax pe r pack from 2005 through 2010. Aerage State Cigarehe Tax per Puck Year Source:"Trends in Average State Cigarette Ta:.: Rates."' Campaign for Tobacco-Free Kids Let 0 11 represent the average state cigarette tax pe r pack, in t~ n years after 2004. a. Use the numbe rs given in the graph to find and interpret 1 6 6 - 2;a,. ,_, b. ll1e finite sequence whose general term is a" - lln + 78, where n - 1, 2, 3, 4, 5, 6, models the average sta te cigarette tax per pack, in cent~ 11 years after 2004. Use this model to find - 6 1 a;. How does this compare with the i- 1 actual value you obtained in part (a)?

A deposil of S6000 is made in an account t hat earns 6% interest compounded quarterly. The balance in the acc.ount after n q uarters is given by the seque nce ( 0.06) a. - 6000 I + 4 , n - 1,2,3 . ... Find the balance i n the account after five years. Round to the nearest cent.

A deposit of $10,000 is made in an account that earns S% interest compounded quarterly. The balance in the account after n q uarters is given by the seque nce ( o.os) a, - 10.000 I + - 4 - , II - I. 2, 3 .. ... Find the balance in the account afte r six years. Round to the nearest cent.

What is a sequence? Give an example with your description.

Explain how to write terms of a sequence if the formula for the general term is given.

What does the graph of a sequence look like? How is it obta ined?

What is a recursion formula?

Explain how to find n! if n is a positive integer.

E . b I 900! . I . xplam the est way lo eva uate 899!' \\'lthout a ca culator.

What is the meaning o f the symboi 'Z? Give an example with your de.'icription.

You buy a new ca r for $24,000. At the end of n yea ~ the value of your car is given by the sequence a, - 24,ooo(f)". n - 1.1, 3, .... Find a5 a nd write a sente nce explaining what this value represents. Describe the nth tenn of the sequence in tenns of the value of your car a t the end of each year.

In Exercises 81-85, use a calculator's factorial key to evaluate each expression. ~ 198!

In Exercises 81-85, use a calculator's factorial key to evaluate each expression. (::}

In Exercises 81-85, use a calculator's factorial key to evaluate each expression. 300 ~

In Exercises 81-85, use a calculator's factorial key to evaluate each expression. 20! (20 3)!

In Exercises 81-85, use a calculator's factorial key to evaluate each expression. 54! (54 3)!3!

Use the I SEQ I (seque nce) ca pability of a graphing utility to verify the tenns of the sequences you obtained for any five sequences from Exercises 1-t2 or 19-22.

Use the !SUM II SEQ I (sum of the sequence) capability of a graphing utility to verify any five of the sums you obtained in Exercises 29-42.

As n increases, the te rms of the sequence a - (1 + .!.) n II get closer and closer to the number e (where e ~e 2.7183). Use a calc.ulato r to find aiO,aloo,a ullll): am.ooo., and aiOO.tllll): comparing these te rms to your calculato r's decimal approximation for e.

Many graphing utilities Jzm1e asequence-graphh1g mode that plots the fenns of a sequen as points on a rectangular coordinate system. Consult your manual; if your graphing ulibly has ihis capability. use it to graph each of the sequena:s in Exercises 89-92. What appears to be happening to the terms of e.ach sequence as n gels larger? a. - ;;-+"j 11: [0, 10, 1] by a. : [0. I, 0.1]

Many graphing utilities Jzm1e asequence-graphh1g mode that plots the fenns of a sequen as points on a rectangular coordinate system. Consult your manual; if your graphing ulibly has ihis capability. use it to graph each of the sequena:s in Exercises 89-92. What appears to be happening to the terms of e.ach sequence as n gels larger? a.- 7 n:[O, 1000, JOO] by a.:[O.J.O.I]

Many graphing utilities Jzm1e asequence-graphh1g mode that plots the fenns of a sequen as points on a rectangular coordinate system. Consult your manual; if your graphing ulibly has ihis capability. use it to graph each of the sequena:s in Exercises 89-92. What appears to be happening to the terms of e.ach sequence as n gels larger? a11 - 3 II 11: (0, 10, I) by a.: (0.2,0.2)

Many graphing utilities Jzm1e asequence-graphh1g mode that plots the fenns of a sequen as points on a rectangular coordinate system. Consult your manual; if your graphing ulibly has ihis capability. use it to graph each of the sequena:s in Exercises 89-92. What appears to be happening to the terms of e.ach sequence as n gels larger? a~~ .1 , 5n +21r+1 n:[O, 10, l]by a.:[O, 1,0.1]

In Exercises 93-96, determine whether each statem~nt makes sense or does not make sense, and explain your reason mg. Now that I've studied sequences, ! realize that the joke in this cartoon is based on the fact that you can't have a negative number of sheep. Harry Blair

In Exercises 93-96, determine whether each statem~nt makes sense or does not make sense, and explain your reason mg. By writing a~. a2 a3, a.;.,, . . , a, , . . . , I can see that the range of a sequence is the set of positive integers.

In Exercises 93-96, determine whether each statem~nt makes sense or does not make sense, and explain your reason mg. It makes a diffe rence whether or not I use parentheses around the expression fo llowing the summation syrnbol, because the value of 2; ' (i + 7) is92, but the value of 2; . i + 7 is 43.

In Exercises 93-96, determine whether each statem~nt makes sense or does not make sense, and explain your reason mg. Without writing out the te rms. I can see that (- 1) 111 in (- 1)'" a11 - - 3 - ca uses the terms to alternate in sign.

In Exercises 97-100, determine whether ~11ch statement is true or false./[ the statement is false, make the necessary changc(s) to produce a true statement. !!L - _!_ (n - t)! n - I

In Exercises 97-100, determine whether ~11ch statement is true or false./[ the statement is false, make the necessary changc(s) to produce a true statement. The Fibonacci sequence I. I, 2. 3. 5, 8. 13, 21. 34. 55, 89. 144 . ... can be defined recursively using a0 - I. a 1 - 1; an - U11 _ 1 + ''"_1, where 11 :a 2.

In Exercises 97-100, determine whether ~11ch statement is true or false./[ the statement is false, make the necessary changc(s) to produce a true statement. 2: (- 1)'2' - 0

In Exercises 97-100, determine whether ~11ch statement is true or false./[ the statement is false, make the necessary changc(s) to produce a true statement. 2;a,b, - 2;a,2;b;

Write Lhe first five temls of the sequence whose first term is 9 and whose general term is a. - {a , 3aa- l - 5 if a a - 1 is odd if a a - 1 is odd for n " 2.

Enough curiosities involving the Fibonacci sequence exist to warrant a Houri.shing Fibonacci Associ3Lion. which publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the Internet or the rcscorch department of your library. and find one property that interests you. After doing this research, get together with your group to shore these intriguing pmperties.

Exercises 103-105 will help you prepare for the material covered in the next section. Consider the sequence 8.3. - 2,- 7.-12 ... .. Finda2 - ua. al - a2, a a - a3. and JJs - aJ.. \Vhal do you observe?

Exercises 103-105 will help you prepare for the material covered in the next section. Conside r the sequence whose tllh term i.s a, - 411 - 3. Find a1 - a 1 a3 - a2 a" - OJ- and a, - a". \\'hat do you obscr\'e?

Exercises 103-105 will help you prepare for the material covered in the next section. Use the formula a. - 4 T (n - 1)(-7) to find the eighth tern> of the sequence 4,-3, - tO . ....