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All Textbook Solutions for College Algebra

For the following exercises, graph the given conic section. hit is a parabola. label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. 26. Find a polar equation of the conic with focus at the origin, eccentricity of e=2 , and directrix: x=3 .Write the first five terms of the sequence defined by the explicit formula tn=5n4 .Write the first five terms of the sequence. an=4n(2)n Write the first six terms of the sequence. an={5n2ifniseven2n3ifnisodd Write an explicit formula for the nth term of the sequence. 9,81,729,6,561,59,049,...Write an explicit formula for the nth term of the sequence. {34,98,2712,8116,24320,...}Write an explicit formula for the nth term of the sequence. {1e2,1e,1,e,e2,...}Write the first five terms of the sequence defined by the recursive formula. a1=2an=2an1+1 for n2 Write the first 8 terms of the sequence defined by the recursive formula. a1=0a2=1a3=1an=an1an2+an3,forn4 Write the first five terms of the sequence defined by the explicit formula an=(n+1)!2n Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?Describe three ways that a sequence can be defined.Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.What happens to the terms an 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?What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.For the following exercises, write the first four terms of the sequence. 6. an=2n2 For the following exercises, write the first four terms of the sequence. an=16n+1 For the following exercises, write the first four terms of the sequence. 8. an=(5)n1 For the following exercises, write the first four terms of the sequence. 9. an=2nn3 For the following exercises, write the first four terms of the sequence. an=2n+1n3 For the following exercises, write the first four terms of the sequence. 11. an=1.25(4)n1 For the following exercises, write the first four terms of the sequence. 12. an=4(6)n1 For the following exercises, write the first four terms of the sequence. an=n22n+1 For the following exercises, write the first four terms of the sequence. a n = ( 10 ) n +1 For the following exercises, write the first four terms of the sequence. 15. an=(4(5)n15)For the following exercises, write the first four terms of the sequence. 16. an={(3)n1ifnisodd(2)n2ifniseven For the following exercises, write the first four terms of the sequence. 17. an={n25ifn5n22n+1ifn5 For the following exercises, write the first four terms of the sequence. 18. an={(2n+1)2ifnisdivisibleby42nifnisnotdivisibleby4 For the following exercises, write the first eight terms of the piecewise sequence. 19. an={2.5(2)n1ifniscomposite0.65n1ifnisprimeor1 For the following exercises, write the first eight terms of the piecewise sequence. 20. an={4(n22)ifn3orn6n224if3n6 For the following exercises, write an explicit formula for each sequence. 21. 4,7,12,19,28,...For the following exercises, write a recursive formula for each sequence. 22. 4,2,10,14,34 For the following exercises, write an explicit formula for each sequence. 23. 1,1,43,2,165,...For the following exercises, write an explicit formula for each sequence. 24. 0,1e11+e2,1e21+e3,1e31+e4,1e41+e5,...For the following exercises, write an explicit formula for each sequence. 112,14,18,116,....For the following exercises, write a recursive formula for each sequence. a1=9,an=an1+n For the following exercises, write the first five terms of the sequence. a1=3,an=(3)an1 For the following exercises, write the first five terms of the sequence. a1=4,an=an1+2nan11 For the following exercises, write the first five terms of the sequence. 29. a1=1,an=(3)n1an12 For the following exercises, write the first five terms of the sequence. 30. a1=30,an=(2+an1)(12)n For the following exercises, write the first eight terms of the sequence. 31. a1=124,a2=1,an=(2an2)(3an1)For the following exercises, write the first eight terms of the sequence. an=1,an=5,an=an2(3an1)For the following exercises, write the first eight terms of the sequence. 33. a1=2,a2=10,an=2(an1+2)an2 For the following exercises, write a recursive formula for each sequence. 34. 2.5,5,10,20,40,... ...For the following exercises, write a recursive formula for each sequence. 35. 8,6,3,1,6,...For the following exercises, write a recursive formula for each sequence. 36. 2,4,12,48,240,...For the following exercises, write a recursive formula for each sequence. 37. 35,38,41,44,47,...For the following exercises, write a recursive formula for each sequence. 38. 15,3,35,325,3125,....For the following exercises, evaluate the factorial. 39. 6!For the following exercises, evaluate the factorial. 40. (126)!For the following exercises, evaluate the factorial. 41. 12!6!For the following exercises, evaluate the factorial. 42. 100!99!For the following exercises, write the first four terms of the sequence. 43. an=n!n2 For the following exercises, write the first four terms of the sequence. 44. an=3n!4n!For the following exercises, write the first four terms of the sequence. 45. an=n!n2n1 For the following exercises, write the first four terms of the sequence. 46. an=100nn(n1)!For the following exercises, graph the first five terms of the indicated sequence 47. an=(1)nn+n For the following exercises, graph the first five terms of the indicated sequence 48. an={3+nifnisodd4+n2nifniseven For the following exercises, graph the first five terms of the indicated sequence 49. a1=2,an=(an1+1)2 For the following exercises, graph the first five terms of the indicated sequence 50. an=1,an=an1+8 For the following exercises, graph the first five terms of the indicated sequence 51. an=(n+1)!(n1)For the following exercises,write an explicit formula for the sequence using the first five points shown on the graph.For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND]ASS for the previous term an1 .. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 57. Find the first terms of the sequence a1=87111 , an=43an1+1237 . Use the > Frac feature to give fractional results.Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ASS for the previous term an1 .. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 58. Find the 15th term of the sequence a1=625,an=0.8an1+18 Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ASS for the previous term an1 .. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 59. Find the first five terms of the sequence a1=2,an=2[(an1)1]+1 Follow these steps to evaluate a sequence defined recursively using a graphing calculator: •On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ASS for the previous term an1 .. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 60. Find the first ten terms of the sequence a1=8,an=(an1+1)!an1!Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ASS for the previous term an1 .. Press [ENTER]. • Continue pressing [ENTER] to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 61. Find the tenth term of the sequence a1=2,an=nan1 Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n• In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value ofnthat ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 62. List the first five terms of the sequence. an=289n+53 Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n• In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of nthat ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 63. List the first six terms of the sequence. an=n33.5n2+4.1n1.52.4n Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n • In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of nthat ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 64. List the first five terms of the sequence. an=15n(2)n147 Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n • In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 65. List the first four terms of the sequence. an=5.7n+0.275(n1)Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n • In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 66. List the first six terms of the sequence an=n!n Consider the sequence defined by an=68n. Is an=421 a term in the sequence? Verify the result.What term in the sequence an=n2+4n+42(n+2) has the value 41? Verify the result.Find a recursive formula for the sequence 1,0,1,1,0,1,1,0,1,1,0,1,1,... (Hint: find a pattern for an based on the first two terms.)Calculate the first eight terms of the sequences an=(n+2)!(n1)! and bn=n3+3n32n , and then make a conjecture about the relationship between these two sequences.Prove the conjecture made in the preceding exercise.Is the given sequence arithmetic? If so, find the common difference. 18,16,14,12,10,...Is the given sequence arithmetic? If so, find the common difference. {1,3,6,10,15,...}List the first five terms of the arithmetic sequence with a1=1andd=5 .Given a3=7 and a5=17 , find a2 .Write a recursive formula for the arithmetic sequence. 25,37,49,61,...Write an explicit formula for the following arithmetic sequence. 50,47,44,41,Find the number of terms in the finite arithmetic sequence. 6,11,16,...,56 A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How tong will her daily run be 8 weeks from today?What is an arithmetic sequence?How is the common difference of an arithmetic sequence found?How do we determine whether a sequence is arithmetic?What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?Describe how linear functions and arithmetic sequences are similar. How are they different?For the following exercises, find the common difference for the arithmetic sequence provided. 6. 5,11,17,23,29,For the following exercises, find the common difference for the arithmetic sequence provided. 7. {0,12,1,32,2,...}For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. 8. {11.4,9.3,7.2,5.1,3,...}For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. 9. 4,16,64,256,1024,...For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. 10. a1=25,d=9 For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. 11. a1=0,d=23 For the following exercises, write the first five terms of the arithmetic series given two terms. 12. a1=17,a7=31 For the following exercises, write the first five terms of the arithmetic series given two terms. 13. a13=60,a33=160 For the following exercises, find the first term given two terms from an arithmetic sequence. 14.First term is 3, common difference is 4, find the5th term.For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. 15. First term is 4, common difference is 5, find the 4th term.For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference 16. First term is 5, common difference is 6, find the 8th term.For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference 17. First term is 6, common difference is 7, find the 6th term.For the following exercises, find the first term given two terms from an arithmetic sequence. 18. First term is 7, common difference is 8, find the 7th term.For the following exercises, find the first term given two terms from an arithmetic sequence. 19. Find the first term or a3 of an arithmetic sequence if a6=12 and a14=28 .For the following exercises, find the first term given two terms from an arithmetic sequence. 20. Find the first term or a1of an arithmetic sequence if a7=21anda15=42 .For the following exercises, find the first term given two terms from an arithmetic sequence. 21. Find the first term or a1 of an arithmetic sequence if a8=40anda23=115 .For the following exercises, find the first term given two terms from an arithmetic sequence. 22. Find the first term or a1 of an arithmetic sequence if a9=54 and a17=102 .For the following exercises, find the first term given two terms from an arithmetic sequence. 23. Find the first term or a1 of an arithmetic sequence if a11=11anda21=16 For the following exercises, find the specified term given two terms from an arithmetic sequence. 24. a1=33anda7=15.Finda4 .For the following exercises, find the specified term given two terms from an arithmetic sequence. 25. a3=17.1anda10=15.7.Finda21 .For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 26. a1=39;an=an13 For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 27. a1=19;an=an11.4 For the following exercises, write an explicit formula for each arithmetic sequence. 28. a={40,60,80,...}For the following exercises, write an explicit formula for each arithmetic sequence. 29. a={17,26,35,...}For the following exercises, write a recursive formula for each arithmetic sequence. 30. a={1,2,5,...}For the following exercises, write a recursive formula for each arithmetic sequence. 31. a=(12,17,22,...For the following exercises, write a recursive formula for each arithmetic sequence. 32. a={15,7,1,...}For the following exercises, write a recursive formula for each arithmetic sequence. 33. a=8.9,10.3,11.7,For the following exercises, write a recursive formula for each arithmetic sequence. 34. a={0.52,1.02,1.52,...}For the following exercises, write a recursive formula for each arithmetic sequence. 35. a={15,920,710,...}For the following exercises, write a recursive formula for each arithmetic sequence. 36. a={12,54,2,...}For the following exercises, write a recursive formula for each arithmetic sequence. 37. a={16,1112,2,...}For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 38. a={7,4,1,...} ; Find the 17thterm.For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 39. a=4,11,18,... ; Find the 14thterm.For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 40. a={2,6,10,...} ; Find the 12th term.For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 41. a=244n For the following exercises, write an explicit formula for each arithmetic sequence. 42. a=12n12 For the following exercises, write an explicit formula for each arithmetic sequence. 43. a={3,5,7,...}For the following exercises, write an explicit formula for each arithmetic sequence. 44. a={32,24,16,...}For the following exercises, write an explicit formula for each arithmetic sequence. 45. a={5,95,195,...}For the following exercises, write an explicit formula for each arithmetic sequence. 46. a={17,217,417,...}For the following exercises, write an explicit formula for each arithmetic sequence. 47. a={1.8,3.6,5.4,...}For the following exercises, write an explicit formula for each arithmetic sequence. 48. a={18.1,16.2,14.3,...}For the following exercises, write an explicit formula for each arithmetic sequence. 49. a={15.8,18.5,21.2,...}For the following exercises, write an explicit formula for each arithmetic sequence. 50. a={13,43,3,...}For the following exercises, write a recursive formula for each arithmetic sequence. 51. a={0,13,23,...}For the following exercises, write a recursive formula for each arithmetic sequence. 52. a={5,103,53,...}For the following exercises, find the number of terms in the given finite arithmetic sequence. 53. a={3,4,11,...,60}For the following exercises, find the number of terms in the given finite arithmetic sequence. 54. a={1.2,1.4,1.6,..,3.8}For the following exercises, find the number of terms in the given finite arithmetic sequence. 55. a={12,2,72,....,8}For the following exercises, determine whether the graph shown represents an arithmetic sequence. 56.For the following exercises, find the first term given two terms from an arithmetic sequence. 57.For the following exercises, find the common difference for the arithmetic sequence provided. a1=0,d=4 For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. a1=9;an=an110 For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. an=12+5n For the following exercises, follow the steps to work with the arithmetic sequence an=3n2 using a graphing calculator: • Press [MODE] Select [SEQ] in the fourth line Select [DOT] in the fifth line Press [ENTER] • Press [Y = ] nMin Is the first counting number for the sequence. Set nMin=1u(n) is the pattern for the sequence. Set u(n)=3n2u(nMin) is the first number in the sequence. Set u(nMin)=1 • Press [2ND] then [WINDOW] to go to TRLSET Set TblStart=1 Set Tbl=l Set lndpnt: Auto and Depend: Auto • Press [2ND] then [GRAPH] to go to the [TABLE] 61. What are the first seven terms shown in the column with the heading u(n)?For the following exercises, follow the steps to work with the arithmetic sequence an=3n2 using a graphing calculator: • Press [MODE] Select [SEQ] in the fourth line Select [DOT] in the fifth line Press [ENTER] • Press [Y = ] nMin Is the first counting number for the sequence. Set nMin=1u(n) is the pattern for the sequence. Set u(n)=3n2u(nMin) is the first number in the sequence. Set u(nMin)=1 • Press [2ND] then [WINDOW] to go to TRLSET Set TblStart=1 Set Tbl=l Set lndpnt: Auto and Depend: Auto • Press [2ND] then [GRAPH] to go to the [TABLE] 62. Use the scroll-down arrow to scroll to n=50 . What value is given for u(n)?For the following exercises, follow the steps to work with the arithmetic sequence an=3n2 using a graphing calculator: • Press [MODE] Select [SEQ] in the fourth line Select [DOT] in the fifth line Press [ENTER] • Press [Y = ] nMin Is the first counting number for the sequence. Set nMin=1u(n) is the pattern for the sequence. Set u(n)=3n2u(nMin) is the first number in the sequence. Set u(nMin)=1 • Press [2ND] then [WINDOW] to go to TRLSET Set TblStart=1 Set Tbl=l Setlndpnt: Auto and Depend: Auto • Press [2ND] then [GRAPH] to go to the [TABLE] 63. Press [WINDOW]. Set nMin=1,nMax=5,xMin=0,xMax=6,yMin=1,andyMax=14 . then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.For the following exercises, follow the steps given above to work with the arithmetic sequence an=12n+5 using a graphing calculator. 64. What are the first seven terms shown in the column with the heading u(n) in the [TABI E] feature?For the following exercises, follow the steps given above to work with the arithmetic sequence an=12n+5 using a graphing calculator. 65. Graph the sequence as it appears on the graphing calculator. Be sure to adjust the [WINDOW] settings as needed.Give two examples of arithmetic sequences whose 4thterms are 9.Give two examples of arithmetic sequences whose 10thterms are 206.Find the 5thterm of the arithmetic sequence 9b,5b,b, .Find the 11th term of the arithmetic sequence 3a2b,a+2b,-a+6b,.... .At which term does the term does the sequence 5.4,14.5,23.6, exceed 151?At which term does the sequence {173,316,143,...} begin to have negative values?For which terms does the finite arithmetic sequence {52,198,94,...,18} have integer va1ues?Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31stterm.Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28thterm.Is the sequence geometric? If so, find the common ratio. 5,10,15,20,...Is the sequence geometric? If so, find the common ratio. 100,20,445,..List the first five terms of the geometric sequence with a1=18 and r=13 .Write a recursive formula for the following geometric sequence. {2,43,89,1627,...}Given a geometric sequence with a2=4 and a3=32 , find a6.Write an explicit formula for the following geometric sequence. {1,3,9,27....}A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will Increase by 2.6% per week. a. Write a formula for the number of hits. b. Estimate the number of hits in 5 weeks.What is a geometric sequence?How is the common ratio of a geometric sequence found?What is the procedure for determining whether a sequence is geometric?What is the difference between an arithmetic sequence and a geometric sequence?Describe how exponential functions and geometric sequences are similar. How are they different?For the following exercises, find the common ratio for the geometric sequence. 6. 1,3,9,27,81,...For the following exercises, find the common ratio for the geometric sequence. 7. 0.125,0.25,0.5,1,2,...For the following exercises, find the common ratio for the geometric sequence. 8. 2,12,18,132,1128,...For the following exercises, find the common ratio for the geometric sequence. 9. 6,12,24,48,96,...For the following exercises, find the common ratio for the geometric sequence. 10. 5,5.2,5.4,5.6,5.8,...For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 11. 1,12,14,18,116,...For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 12. 6,8,11,15,20,...For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 13. 0.8,4,20,100,500 For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 14. a1=8,r=0.3 For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 15. a1=5,r=15 For the following exercises, write the first five terms of the geometric sequence, given any two terms. 16. a7=64,a10=512 For the following exercises, write the first five terms of the geometric sequence, given any two terms. a6=25,a8=6.25 For the following exercises, find the specified term for the geometric sequence. given the first term and common ratio. The first term is 2, and the common ratio is 3. Find the 5thterm.For the following exercises, find the specified term for the geometric sequence. given the first term and common ratio. The first term is 16 and the common ratio is 13. Find the 4th term.For the following exercises, find the specified term for the geometric sequence, given the first four terms. an={1,2,4,8,...}. Find a12 .For the following exercises, find the specified term for the geometric sequence, given the first four terms. a n ={ 2, 2 3 , 2 9 , 2 27 ,.... }Find a 7 For the following exercises, write the first five terms of the geometric sequence. a1=486,an=13an1 For the following exercises, write the first five terms of the geometric sequence. 23. a1=7,an=0.2an1 For the following exercises, write the first five terms of the geometric sequence. an={1,5,25,125,...}For the following exercises, write the first five terms of the geometric sequence. an={32,16,8,4,...}For the following exercises, write the first five terms of the geometric sequence. an={14,56,224,896,...}For the following exercises, write the first five terms of the geometric sequence. an={10,3,0.9,0.27,...}For the following exercises, write the first five terms of the geometric sequence. 28. an={0.61,1.83,5.49,16,47,...}For the following exercises, write the first five terms of the geometric sequence. 29. an={35,110,160,1360,....}For the following exercises, write the first five terms of the geometric sequence. an={2,43,89,1627,...}For the following exercises, write the first five terms of the geometric sequence. 31. an={,1512,1128,132,18,...}For the following exercises, write the first five terms of the geometric sequence. 32. an=45n1 For the following exercises, write the first five terms of the geometric sequence. 33. an=12(12)n1 For the following exercises, write the first five terms of the geometric sequence. 34. an={2,4,8,16,...}For the following exercises, write the first five terms of the geometric sequence. 35. an={1,3,9,27,...}For the following exercises, write the first five terms of the geometric sequence. 36. an={4,12,36,108,...}For the following exercises, write the first five terms of the geometric sequence. 37. an={0.8,4,20,100,...}For the following exercises, write an explicit formula for each geometric sequence. 38. an={1,25,5,20,80,...}For the following exercises, write an explicit formula for each geometric sequence. 39. a={1,45,1625,64125,...}For the following exercises, write an explicit formula for each geometric sequence. 40. an={2,13,118,1108,...}For the following exercises, write an explicit formula for each geometric sequence. 41. an={3,1,13,19,...}For the following exercises, find the specified term for the geometric sequence given. 42. Let a1=4,an=3an1 .Find a8 .For the following exercises, find the specified term for the geometric sequence given. 43. Let an=(13)n1.Finda12 For the following exercises, write an explicit formula for each geometric sequence. 44. an={1,3,9,...,2187}For the following exercises, write an explicit formula for each geometric sequence. 45. an={2,1,12,...,11024}For the following exercises, determine whether the graph shown represents a geometric sequence.For the following exercises, determine whether the graph shown represents a geometric sequence.For the following exercises, use the information provided to graph the first five terms of the geometric sequence. a1=1,r=12 For the following exercises, use the information provided to graph the first five terms of the geometric sequence. 49. a1=3,an=2an1 For the following exercises, use the information provided to graph the first five terms of the geometric sequence. 50. an=270.3n1 Use recursive formulas to give two examples of geometric sequences whose 3rd terms are 200.Use explicit formulas to give two examples of geometric sequences whose 7thterms are 1024.Find the 5thterm of the geometric sequence {b,4b,16b,...}.Find the 7th term of the geometric sequence 64a(b),32a(3b),16a(9b),... .At which term does the sequence 10,12,14.4,17.28,... exceed 100?At which term does the sequence {12187,1729,1243,181....} begin to have integer values?For which term does the geometric sequence an=36(23)n1 first have a non-integer value?Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10thterm.Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8th term.Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.Eva1uate k=25(3k1) .Use the formula to find the sum of the arithmetic series. 1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4 Use the formula to find the sum of the arithmetic series. 13+21+29++69 Use the formula to find the sum of the arithmetic series. k=11056k A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?Use the formula to find the indicated partial sum of each geometric series. S20 for the series 1,000+500+250+...Use the formula to find the indicated partial sum of each geometric series. k=183k At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years’Determine whether the sum of the infinite series is defined. 13+12+34+98+...Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+Determine whether the sum of the infinite series is defined. k=115(0.3)k Find the sum, if it exists. 2+23+29+...Find the sum, if it exists. k=10.76k+1 Find the sum, if it exists. k=1(38)k At the beginning of each month. $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?What is an nth partial sum?What is the difference between an arithmetic sequence and an arithmetic series?What is a geometric series?How is finding the sum of an infinite geometric series different from finding the nth partial sum?What is an annuity?For the following exercises, express each description of a sum using summation notation. 6. The sum of terms m2+3m from m=1tom=5 For the following exercises, express each description of a sum using summation notation. 7. The sum from of n=0ton=4of5n For the following exercises, express each description of a sum using summation notation. 8.The sum of 6k5 from k=2tok=1 For the following exercises, express each description of a sum using summation notation. 9. The sum that results from adding the number 4 five times For the following exercises, express each arithmetic sum using summation notation. 10. 5+10+15+20+25+30+35+40+45+50 For the following exercises, express each arithmetic sum using summation notation. 11. 10+18+26+...+162 For the following exercises, express each arithmetic sum using summation notation. 12. 12+1+32+2+...+4 For the following exercises, use the formula for the sum of the first n terms of each arithmetic sequence. 13. 32+2+52+3+72 For the following exercises, use the formula for the sum of the first n terms of each arithmetic sequence. 14. 19+25+31+...+73 For the following exercises, use the formula for the sum of the first n terms of each arithmetic sequence. 15. 3.2+3.4+3.6+...+5.6 For the following exercises, express each geometric sum using summation notation. 16. 1+3+9+27+81+243+729+2187 For the following exercises, express each geometric sum using summation notation. 17. 8+4+2+...+0.125 For the following exercises, express each geometric sum using summation notation. 18. 16+112124+...+1768 For the following exercises, use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. 19. 9+3+1+13+19 For the following exercises, use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. 20. n=1952n1 For the following exercises, use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. 21. a=111640.2a1 For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. 22. 12+18+24+30+...For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. 23. 2+1.6+1.28+1.024+...For the following exercise, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. 24. m=14m1 For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. 25. k=1(12)k1 For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20. 26. Graph the arithmetic sequence showing one year of Javier’s deposits.For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20. 27. Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.For the following exercises, use the geometric series k=1(12)k 28. Graph the first 7 partial sums of the series.For the following exercises, use the geometric series k=1(12)k 29. What number does Sn seem to be approaching in the graph? Find the sum to explain why this makes sense.For the following exercises, find the indicated sum. 30. a=114a For the following exercises, find the indicated sum. 31. n=16n(n2)For the following exercises, find the indicated sum. 32. k=117k2 For the following exercises, find the indicated sum. 33. k=172k For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. 34. 1.7+0.4+0.9+2.2+3.5+4.8 For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. 35. 6+152+9+212+12+272+15 For the following exercises, use the formula for the sum of the first n terms of an arithmetic series to find the sum. 36. 1+3+7+...+31 For the following exercises, use the formula for the sum of the first n terms of an arithmetic series to find the sum. 37. k=111(k212)For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. 38. S6 for the series 21050250...For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. 39. S7 For the series 0.42+1050 For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. k=192k1 For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. n=192.( 1 2)n1 For the following exercises, find the sum of the infinite geometric series. 4+2+1+12...For the following exercises, find the sum of the infinite geometric series. 1 1 4 1 16 1 64 ....For the following exercises, find the sum of the infinite geometric series. n=1k=13.( 1 4)k1 For the following exercises, find the sum of the infinite geometric series. 45. n=14.60.5n1 For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. 46. Deposit amount: $50; total deposits: 60 interest rate: 5%, compounded monthly For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. 47. Deposit amount: $150; total deposits: 24; interest rate: 3%, compounded monthly For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. 48. Deposit amount: $450; total deposits 60, interest rate 4.5%, compounded quarterly For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. 49. Deposit amount: $100; total deposits: 120; interest rate: 10%, compounded semi-annually The sum of terms 50k2 from k=x through 7 is 115. What is x?Write an explicit formula for a such that k=06ak=189 . Assume this is an arithmetic series.Find the smallest value of n such that k=1n(3k5)100 How many terms must be added before the series 1357.... has a sum less than -75?Write 0.65 as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert 0.65 to a fraction.The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

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