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Study Guides > College Algebra

Section Exercises

1. Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence? 2. Describe three ways that a sequence can be defined. 3. Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not. 4. What happens to the terms an{a}_{n} of a sequence when there is a negative factor in the formula that is raised to a power that includes n?n? What is the term used to describe this phenomenon? 5. 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{a}_{n}={2}^{n}-2 7. an=16n+1{a}_{n}=-\frac{16}{n+1} 8. an=(5)n1{a}_{n}=-{\left(-5\right)}^{n - 1} 9. an=2nn3{a}_{n}=\frac{{2}^{n}}{{n}^{3}} 10. an=2n+1n3{a}_{n}=\frac{2n+1}{{n}^{3}} 11. an=1.25(4)n1{a}_{n}=1.25\cdot {\left(-4\right)}^{n - 1} 12. an=4(6)n1{a}_{n}=-4\cdot {\left(-6\right)}^{n - 1} 13. an=n22n+1{a}_{n}=\frac{{n}^{2}}{2n+1} 14. an=(10)n+1{a}_{n}={\left(-10\right)}^{n}+1 15. an=(4(5)n15){a}_{n}=-\left(\frac{4\cdot {\left(-5\right)}^{n - 1}}{5}\right) For the following exercises, write the first eight terms of the piecewise sequence. 16. an={(2)n2if n is even(2)n1if n is odd{a}_{n}=\begin{cases}\left(−2\right)^{n}−2 \hfill& \text{if }n\text{ is even} \\ \left(2\right)^{n−1} \hfill& \text{if }n\text{ is odd}\end{cases} 17. an={n22n+1if n5n25if n>5{a}_{n}=\begin{cases}\frac{n^{2}}{2n+1}\hfill& \text{if }n\le5 \\ n^{2}−5 \hfill& \text{if }n>5\end{cases} 18. an={(2n+1)2if n is divisible by 42nif n is not divisible by 4{a}_{n}=\begin{cases}\left(2n+1\right)^{2}\hfill& \text{if }n\text{ is divisible by }4 \\ \frac{2}{n}\hfill& \text{if }n\text{ is not divisible by }4\end{cases} 19. an={0.6×5n1if n is prime or 12.5×(2)n1if n is composite{a}_{n}=\begin{cases}−0.6\times5^{n−1}\hfill& \text{if }n\text{ is prime or }1 \\ 2.5\times\left(−2\right)^{n−1}\hfill& \text{if }n\text{ is composite}\end{cases} 20. an={4(n22) if n3 or n>6n224 if 3<n6{a}_{n}=\begin{cases}4\left({n}^{2}-2\right)&\text{ if }{ n }\le{3}\text{ or }{ n }>{ 6 }\\\frac{{n}^{2}-2}{4} & \text{ if }{ 3 }<{ n }\le{ 6 }\end{cases} For the following exercises, write an explicit formula for each sequence. 21. 4,7,12,19,28,4, 7, 12, 19, 28,\dots 22. 4,2,10,14,34,-4,2,-10,14,-34,\dots 23. 1,1,43,2,165,1,1,\frac{4}{3},2,\frac{16}{5},\dots 24. 0,1e11+e2,1e21+e3,1e31+e4,1e41+e5,0,\frac{1-{e}^{1}}{1+{e}^{2}},\frac{1-{e}^{2}}{1+{e}^{3}},\frac{1-{e}^{3}}{1+{e}^{4}},\frac{1-{e}^{4}}{1+{e}^{5}},\dots 25. 1,12,14,18,116,1,-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},\dots For the following exercises, write the first five terms of the sequence. 26. a1=9, an=an1+n{a}_{1}=9,\text{ }{a}_{n}={a}_{n - 1}+n 27. a1=3, an=(3)an1{a}_{1}=3,\text{ }{a}_{n}=\left(-3\right){a}_{n - 1} 28. a1=4, an=an1+2nan11{a}_{1}=-4,\text{ }{a}_{n}=\frac{{a}_{n - 1}+2n}{{a}_{n - 1}-1} 29. a1=1, an=(3)n1an12{a}_{1}=-1,\text{ }{a}_{n}=\frac{{\left(-3\right)}^{n - 1}}{{a}_{n - 1}-2} 30. a1=30, an=(2+an1)(12)n{a}_{1}=-30,\text{ }{a}_{n}=\left(2+{a}_{n - 1}\right){\left(\frac{1}{2}\right)}^{n} For the following exercises, write the first eight terms of the sequence. 31. a1=124, a2=1, an=(2an2)(3an1){a}_{1}=\frac{1}{24},{\text{ a}}_{2}=1,\text{ }{a}_{n}=\left(2{a}_{n - 2}\right)\left(3{a}_{n - 1}\right) 32. a1=1, a2=5, an=an2(3an1){a}_{1}=-1,{\text{ a}}_{2}=5,\text{ }{a}_{n}={a}_{n - 2}\left(3-{a}_{n - 1}\right) 33. a1=2, a2=10, an=2(an1+2)an2{a}_{1}=2,{\text{ a}}_{2}=10,\text{ }{a}_{n}=\frac{2\left({a}_{n - 1}+2\right)}{{a}_{n - 2}} For the following exercises, write a recursive formula for each sequence. 34. 2.5,5,10,20,40,-2.5,-5,-10,-20,-40,\dots 35. 8,6,3,1,6,-8,-6,-3,1,6,\dots 36. 2, 4, 12, 48, 240, 2,\text{ }4,\text{ }12,\text{ }48,\text{ }240,\text{ }\dots 37. 35, 38, 41, 44, 47, 35,\text{ }38,\text{ }41,\text{ }44,\text{ }47,\text{ }\dots 38. 15,3,35,325,3125,15,3,\frac{3}{5},\frac{3}{25},\frac{3}{125},\cdots For the following exercises, evaluate the factorial. 39. 6!6! 40. (126)!\left(\frac{12}{6}\right)! 41. 12!6!\frac{12!}{6!} 42. 100!99!\frac{100!}{99!} For the following exercises, write the first four terms of the sequence. 43. an=n!n2{a}_{n}=\frac{n!}{{n}^{\text{2}}} 44. an=3n!4n!{a}_{n}=\frac{3\cdot n!}{4\cdot n!} 45. an=n!n2n1{a}_{n}=\frac{n!}{{n}^{2}-n - 1} 46. an=100nn(n1)!{a}_{n}=\frac{100\cdot n}{n\left(n - 1\right)!} For the following exercises, graph the first five terms of the indicated sequence 47. an=(1)nn+n{a}_{n}=\frac{{\left(-1\right)}^{n}}{n}+n 48. an={4+n2nif n is even3+nif n is odd{a}_{n}=\begin{cases}\frac{4+n}{2n} & \text{if }n\text{ is even} \\ 3+n & \text{if }n\text{ is odd}\end{cases} 49. a1=2, an=(an1+1)2{a}_{1}=2,\text{ }{a}_{n}={\left(-{a}_{n - 1}+1\right)}^{2} 50. an=1, an=an1+8{a}_{n}=1,\text{ }{a}_{n}={a}_{n - 1}+8 51. an=(n+1)!(n1)!{a}_{n}=\frac{\left(n+1\right)!}{\left(n - 1\right)!} For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph. 52. Graph of a scattered plot with labeled points: (1, 5), (2, 7), (3, 9), (4, 11), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n. 53. Graph of a scattered plot with labeled points: (1, 0.5), (2, 1), (3, 2), (4, 4), and (5, 8). The x-axis is labeled n and the y-axis is labeled a_n. 54. Graph of a scattered plot with labeled points: (1, 12), (2, 9), (3, 6), (4, 3), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n. For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph. 55. Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n. 56. Graph of a scattered plot with labeled points: (1, 16), (2, 8), (3, 4), (4, 2), and (5, 1). The x-axis is labeled n and the y-axis is labeled a_n. 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{a}_{1} and press [ENTER].
  • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term an1{a}_{n - 1}. 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 five terms of the sequence a1=87111, an=43an1+1237{a}_{1}=\frac{87}{111},\text{ }{a}_{n}=\frac{4}{3}{a}_{n - 1}+\frac{12}{37}. Use the >Frac feature to give fractional results. 58. Find the 15th term of the sequence a1=625,an=0.8an1+18{a}_{1}=625, {a}_{n}=0.8{a}_{n - 1}+18. 59. Find the first five terms of the sequence a1=2,an=2[(an1)1]+1{a}_{1}=2, {a}_{n}={2}^{\left[\left({a}_{n}-1\right)-1\right]}+1. 60. Find the first ten terms of the sequence a1=8, an=(an1+1)!an1!{a}_{1}=8,\text{ }{a}_{n}=\frac{\left({a}_{n - 1}+1\right)!}{{a}_{n - 1}!}. 61. Find the tenth term of the sequence a1=2, an=nan1{a}_{1}=2,\text{ }{a}_{n}=n{a}_{n - 1} Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-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,θ,n]\left[\text{X,T},\theta ,n\right] button for nn
  • In the line headed "Variable:" type in the variable used on the previous step.
  • In the line headed "start:" key in the value of nn that begins the sequence.
  • In the line headed "end:" key in the value of nn 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. 62. List the first five terms of the sequence an=289n+53{a}_{n}=-\frac{28}{9}n+\frac{5}{3}. 63. List the first six terms of the sequence an=n33.5n2+4.1n1.52.4n{a}_{n}=\frac{{n}^{3}-3.5{n}^{2}+ 4.1n - 1.5}{2.4n}. 64. List the first five terms of the sequence an=15n(2)n147{a}_{n}=\frac{15n\cdot {\left(-2\right)}^{n - 1}}{47} 65. List the first four terms of the sequence an=5.7n+0.275(n1)!{a}_{n}={5.7}^{n}+0.275\left(n - 1\right)! 66. List the first six terms of the sequence an=n!n{a}_{n}=\frac{n!}{n}. 67. Consider the sequence defined by an=68n{a}_{n}=-6 - 8n. Is an=421{a}_{n}=-421 a term in the sequence? Verify the result. 68. What term in the sequence an=n2+4n+42(n+2){a}_{n}=\frac{{n}^{2}+4n+4}{2\left(n+2\right)} has the value 41?41? Verify the result. 69. Find a recursive formula for the sequence 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, ... 1,\text{ }0,\text{ }-1,\text{ }-1,\text{ }0,\text{ }1,\text{ }1,\text{ }0,\text{ }-1,\text{ }-1,\text{ }0,\text{ }1,\text{ }1,\text{ }...\text{ }. (Hint: find a pattern for an{a}_{n} based on the first two terms.) 70. Calculate the first eight terms of the sequences an=(n+2)!(n1)!{a}_{n}=\frac{\left(n+2\right)!}{\left(n - 1\right)!} and bn=n3+3n2+2n{b}_{n}={n}^{3}+3{n}^{2}+2n, and then make a conjecture about the relationship between these two sequences. 71. Prove the conjecture made in the preceding exercise.

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