{"id":15638,"date":"2019-09-05T17:18:47","date_gmt":"2019-09-05T17:18:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15638"},"modified":"2019-09-09T21:30:39","modified_gmt":"2019-09-09T21:30:39","slug":"problem-set-39-geometric-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-39-geometric-sequences\/","title":{"raw":"Problem Set 39: Geometric Sequences","rendered":"Problem Set 39: Geometric Sequences"},"content":{"raw":"1. What is a geometric sequence?\r\n\r\n2.\u00a0How is the common ratio of a geometric sequence found?\r\n\r\n3. What is the procedure for determining whether a sequence is geometric?\r\n\r\n4.\u00a0What is the difference between an arithmetic sequence and a geometric sequence?\r\n\r\n5. Describe how exponential functions and geometric sequences are similar. How are they different?\r\n\r\nFor the following exercises, find the common ratio for the geometric sequence.\r\n\r\n6. [latex]1,3,9,27,81,..[\/latex].\r\n\r\n7. [latex]-0.125,0.25,-0.5,1,-2,..[\/latex].\r\n\r\n8.\u00a0[latex]-2,-\\frac{1}{2},-\\frac{1}{8},-\\frac{1}{32},-\\frac{1}{128},..[\/latex].\r\n\r\nFor the following exercises, determine whether the sequence is geometric. If so, find the common ratio.\r\n\r\n9. [latex]-6,-12,-24,-48,-96,..[\/latex].\r\n\r\n10.\u00a0[latex]5,5.2,5.4,5.6,5.8,..[\/latex].\r\n\r\n11. [latex]-1,\\frac{1}{2},-\\frac{1}{4},\\frac{1}{8},-\\frac{1}{16},..[\/latex].\r\n\r\n12.\u00a0[latex]6,8,11,15,20,..[\/latex].\r\n\r\n13. [latex]0.8,4,20,100,500,..[\/latex].\r\n\r\nFor the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.\r\n\r\n14. [latex]\\begin{array}{cc}{a}_{1}=8,& r=0.3\\end{array}[\/latex]\r\n\r\n15. [latex]\\begin{array}{cc}{a}_{1}=5,& r=\\frac{1}{5}\\end{array}[\/latex]\r\n\r\nFor the following exercises, write the first five terms of the geometric sequence, given any two terms.\r\n\r\n16. [latex]\\begin{array}{cc}{a}_{7}=64,& {a}_{10}\\end{array}=512[\/latex]\r\n\r\n17. [latex]\\begin{array}{cc}{a}_{6}=25,& {a}_{8}\\end{array}=6.25[\/latex]\r\n\r\nFor the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.\r\n\r\n18. The first term is [latex]2[\/latex], and the common ratio is [latex]3[\/latex]. Find the 5th<\/sup> term.\r\n\r\n19. The first term is 16 and the common ratio is [latex]-\\frac{1}{3}[\/latex]. Find the 4th<\/sup> term.\r\n\r\nFor the following exercises, find the specified term for the geometric sequence, given the first four terms.\r\n\r\n20. [latex]{a}_{n}=\\left\\{-1,2,-4,8,...\\right\\}[\/latex]. Find [latex]{a}_{12}[\/latex].\r\n\r\n21. [latex]{a}_{n}=\\left\\{-2,\\frac{2}{3},-\\frac{2}{9},\\frac{2}{27},...\\right\\}[\/latex]. Find [latex]{a}_{7}[\/latex].\r\n\r\nFor the following exercises, write the first five terms of the geometric sequence.\r\n\r\n22. [latex]\\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\\frac{1}{3}\\end{array}{a}_{n - 1}[\/latex]\r\n\r\n23. [latex]\\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\\end{array}[\/latex]\r\n\r\nFor the following exercises, write a recursive formula for each geometric sequence.\r\n\r\n24. [latex]{a}_{n}=\\left\\{-1,5,-25,125,...\\right\\}[\/latex]\r\n\r\n25. [latex]{a}_{n}=\\left\\{-32,-16,-8,-4,...\\right\\}[\/latex]\r\n\r\n26.\u00a0[latex]{a}_{n}=\\left\\{14,56,224,896,...\\right\\}[\/latex]\r\n\r\n27. [latex]{a}_{n}=\\left\\{10,-3,0.9,-0.27,...\\right\\}[\/latex]\r\n\r\n28.\u00a0[latex]{a}_{n}=\\left\\{0.61,1.83,5.49,16.47,...\\right\\}[\/latex]\r\n\r\n29. [latex]{a}_{n}=\\left\\{\\frac{3}{5},\\frac{1}{10},\\frac{1}{60},\\frac{1}{360},...\\right\\}[\/latex]\r\n\r\n30.\u00a0[latex]{a}_{n}=\\left\\{-2,\\frac{4}{3},-\\frac{8}{9},\\frac{16}{27},...\\right\\}[\/latex]\r\n\r\n31. [latex]{a}_{n}=\\left\\{\\frac{1}{512},-\\frac{1}{128},\\frac{1}{32},-\\frac{1}{8},...\\right\\}[\/latex]\r\n\r\nFor the following exercises, write the first five terms of the geometric sequence.\r\n\r\n32. [latex]{a}_{n}=-4\\cdot {5}^{n - 1}[\/latex]\r\n\r\n33. [latex]{a}_{n}=12\\cdot {\\left(-\\frac{1}{2}\\right)}^{n - 1}[\/latex]\r\n\r\nFor the following exercises, write an explicit formula for each geometric sequence.\r\n\r\n34. [latex]{a}_{n}=\\left\\{-2,-4,-8,-16,...\\right\\}[\/latex]\r\n\r\n35. [latex]{a}_{n}=\\left\\{1,3,9,27,...\\right\\}[\/latex]\r\n\r\n36.\u00a0[latex]{a}_{n}=\\left\\{-4,-12,-36,-108,...\\right\\}[\/latex]\r\n\r\n37. [latex]{a}_{n}=\\left\\{0.8,-4,20,-100,...\\right\\}[\/latex]\r\n\r\n38.\u00a0[latex]{a}_{n}=\\left\\{-1.25,-5,-20,-80,...\\right\\}[\/latex]\r\n\r\n39. [latex]{a}_{n}=\\left\\{-1,-\\frac{4}{5},-\\frac{16}{25},-\\frac{64}{125},...\\right\\}[\/latex]\r\n\r\n40.\u00a0[latex]{a}_{n}=\\left\\{2,\\frac{1}{3},\\frac{1}{18},\\frac{1}{108},...\\right\\}[\/latex]\r\n\r\n41. [latex]{a}_{n}=\\left\\{3,-1,\\frac{1}{3},-\\frac{1}{9},...\\right\\}[\/latex]\r\n\r\nFor the following exercises, find the specified term for the geometric sequence given.\r\n\r\n42. Let [latex]{a}_{1}=4[\/latex], [latex]{a}_{n}=-3{a}_{n - 1}[\/latex]. Find [latex]{a}_{8}[\/latex].\r\n\r\n43. Let [latex]{a}_{n}=-{\\left(-\\frac{1}{3}\\right)}^{n - 1}[\/latex]. Find [latex]{a}_{12}[\/latex].\r\n\r\nFor the following exercises, find the number of terms in the given finite geometric sequence.\r\n\r\n44. [latex]{a}_{n}=\\left\\{-1,3,-9,...,2187\\right\\}[\/latex]\r\n\r\n45. [latex]{a}_{n}=\\left\\{2,1,\\frac{1}{2},...,\\frac{1}{1024}\\right\\}[\/latex]\r\n\r\nFor the following exercises, determine whether the graph shown represents a geometric sequence.\r\n\r\n46.\r\n\"Graph\r\n\r\n47.\r\n\"Graph\r\n\r\nFor the following exercises, use the information provided to graph the first five terms of the geometric sequence.\r\n\r\n48. [latex]\\begin{array}{cc}{a}_{1}=1,& r=\\frac{1}{2}\\end{array}[\/latex]\r\n\r\n49. [latex]\\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n - 1}\\end{array}[\/latex]\r\n\r\n50. [latex]{a}_{n}=27\\cdot {0.3}^{n - 1}[\/latex]\r\n\r\n51. Use recursive formulas to give two examples of geometric sequences whose 3rd<\/sup> terms are [latex]200[\/latex].\r\n\r\n52.\u00a0Use explicit formulas to give two examples of geometric sequences whose 7th<\/sup> terms are [latex]1024[\/latex].\r\n\r\n53. Find the 5th<\/sup> term of the geometric sequence [latex]\\left\\{b,4b,16b,...\\right\\}[\/latex].\r\n\r\n54.\u00a0Find the 7th<\/sup> term of the geometric sequence [latex]\\left\\{64a\\left(-b\\right),32a\\left(-3b\\right),16a\\left(-9b\\right),...\\right\\}[\/latex].\r\n\r\n55. At which term does the sequence [latex]\\left\\{10,12,14.4,17.28,\\text{ }...\\right\\}[\/latex] exceed [latex]100?[\/latex]\r\n\r\n56.\u00a0At which term does the sequence [latex]\\left\\{\\frac{1}{2187},\\frac{1}{729},\\frac{1}{243},\\frac{1}{81}\\text{ }...\\right\\}[\/latex] begin to have integer values?\r\n\r\n57. For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\\left(\\frac{2}{3}\\right)}^{n - 1}[\/latex] first have a non-integer value?\r\n\r\n58.\u00a0Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10th<\/sup> term.\r\n\r\n59. 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<\/sup> term.\r\n\r\n60.\u00a0Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.","rendered":"

1. What is a geometric sequence?<\/p>\n

2.\u00a0How is the common ratio of a geometric sequence found?<\/p>\n

3. What is the procedure for determining whether a sequence is geometric?<\/p>\n

4.\u00a0What is the difference between an arithmetic sequence and a geometric sequence?<\/p>\n

5. Describe how exponential functions and geometric sequences are similar. How are they different?<\/p>\n

For the following exercises, find the common ratio for the geometric sequence.<\/p>\n

6. [latex]1,3,9,27,81,..[\/latex].<\/p>\n

7. [latex]-0.125,0.25,-0.5,1,-2,..[\/latex].<\/p>\n

8.\u00a0[latex]-2,-\\frac{1}{2},-\\frac{1}{8},-\\frac{1}{32},-\\frac{1}{128},..[\/latex].<\/p>\n

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.<\/p>\n

9. [latex]-6,-12,-24,-48,-96,..[\/latex].<\/p>\n

10.\u00a0[latex]5,5.2,5.4,5.6,5.8,..[\/latex].<\/p>\n

11. [latex]-1,\\frac{1}{2},-\\frac{1}{4},\\frac{1}{8},-\\frac{1}{16},..[\/latex].<\/p>\n

12.\u00a0[latex]6,8,11,15,20,..[\/latex].<\/p>\n

13. [latex]0.8,4,20,100,500,..[\/latex].<\/p>\n

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.<\/p>\n

14. [latex]\\begin{array}{cc}{a}_{1}=8,& r=0.3\\end{array}[\/latex]<\/p>\n

15. [latex]\\begin{array}{cc}{a}_{1}=5,& r=\\frac{1}{5}\\end{array}[\/latex]<\/p>\n

For the following exercises, write the first five terms of the geometric sequence, given any two terms.<\/p>\n

16. [latex]\\begin{array}{cc}{a}_{7}=64,& {a}_{10}\\end{array}=512[\/latex]<\/p>\n

17. [latex]\\begin{array}{cc}{a}_{6}=25,& {a}_{8}\\end{array}=6.25[\/latex]<\/p>\n

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.<\/p>\n

18. The first term is [latex]2[\/latex], and the common ratio is [latex]3[\/latex]. Find the 5th<\/sup> term.<\/p>\n

19. The first term is 16 and the common ratio is [latex]-\\frac{1}{3}[\/latex]. Find the 4th<\/sup> term.<\/p>\n

For the following exercises, find the specified term for the geometric sequence, given the first four terms.<\/p>\n

20. [latex]{a}_{n}=\\left\\{-1,2,-4,8,...\\right\\}[\/latex]. Find [latex]{a}_{12}[\/latex].<\/p>\n

21. [latex]{a}_{n}=\\left\\{-2,\\frac{2}{3},-\\frac{2}{9},\\frac{2}{27},...\\right\\}[\/latex]. Find [latex]{a}_{7}[\/latex].<\/p>\n

For the following exercises, write the first five terms of the geometric sequence.<\/p>\n

22. [latex]\\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\\frac{1}{3}\\end{array}{a}_{n - 1}[\/latex]<\/p>\n

23. [latex]\\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\\end{array}[\/latex]<\/p>\n

For the following exercises, write a recursive formula for each geometric sequence.<\/p>\n

24. [latex]{a}_{n}=\\left\\{-1,5,-25,125,...\\right\\}[\/latex]<\/p>\n

25. [latex]{a}_{n}=\\left\\{-32,-16,-8,-4,...\\right\\}[\/latex]<\/p>\n

26.\u00a0[latex]{a}_{n}=\\left\\{14,56,224,896,...\\right\\}[\/latex]<\/p>\n

27. [latex]{a}_{n}=\\left\\{10,-3,0.9,-0.27,...\\right\\}[\/latex]<\/p>\n

28.\u00a0[latex]{a}_{n}=\\left\\{0.61,1.83,5.49,16.47,...\\right\\}[\/latex]<\/p>\n

29. [latex]{a}_{n}=\\left\\{\\frac{3}{5},\\frac{1}{10},\\frac{1}{60},\\frac{1}{360},...\\right\\}[\/latex]<\/p>\n

30.\u00a0[latex]{a}_{n}=\\left\\{-2,\\frac{4}{3},-\\frac{8}{9},\\frac{16}{27},...\\right\\}[\/latex]<\/p>\n

31. [latex]{a}_{n}=\\left\\{\\frac{1}{512},-\\frac{1}{128},\\frac{1}{32},-\\frac{1}{8},...\\right\\}[\/latex]<\/p>\n

For the following exercises, write the first five terms of the geometric sequence.<\/p>\n

32. [latex]{a}_{n}=-4\\cdot {5}^{n - 1}[\/latex]<\/p>\n

33. [latex]{a}_{n}=12\\cdot {\\left(-\\frac{1}{2}\\right)}^{n - 1}[\/latex]<\/p>\n

For the following exercises, write an explicit formula for each geometric sequence.<\/p>\n

34. [latex]{a}_{n}=\\left\\{-2,-4,-8,-16,...\\right\\}[\/latex]<\/p>\n

35. [latex]{a}_{n}=\\left\\{1,3,9,27,...\\right\\}[\/latex]<\/p>\n

36.\u00a0[latex]{a}_{n}=\\left\\{-4,-12,-36,-108,...\\right\\}[\/latex]<\/p>\n

37. [latex]{a}_{n}=\\left\\{0.8,-4,20,-100,...\\right\\}[\/latex]<\/p>\n

38.\u00a0[latex]{a}_{n}=\\left\\{-1.25,-5,-20,-80,...\\right\\}[\/latex]<\/p>\n

39. [latex]{a}_{n}=\\left\\{-1,-\\frac{4}{5},-\\frac{16}{25},-\\frac{64}{125},...\\right\\}[\/latex]<\/p>\n

40.\u00a0[latex]{a}_{n}=\\left\\{2,\\frac{1}{3},\\frac{1}{18},\\frac{1}{108},...\\right\\}[\/latex]<\/p>\n

41. [latex]{a}_{n}=\\left\\{3,-1,\\frac{1}{3},-\\frac{1}{9},...\\right\\}[\/latex]<\/p>\n

For the following exercises, find the specified term for the geometric sequence given.<\/p>\n

42. Let [latex]{a}_{1}=4[\/latex], [latex]{a}_{n}=-3{a}_{n - 1}[\/latex]. Find [latex]{a}_{8}[\/latex].<\/p>\n

43. Let [latex]{a}_{n}=-{\\left(-\\frac{1}{3}\\right)}^{n - 1}[\/latex]. Find [latex]{a}_{12}[\/latex].<\/p>\n

For the following exercises, find the number of terms in the given finite geometric sequence.<\/p>\n

44. [latex]{a}_{n}=\\left\\{-1,3,-9,...,2187\\right\\}[\/latex]<\/p>\n

45. [latex]{a}_{n}=\\left\\{2,1,\\frac{1}{2},...,\\frac{1}{1024}\\right\\}[\/latex]<\/p>\n

For the following exercises, determine whether the graph shown represents a geometric sequence.<\/p>\n

46.
\n\"Graph<\/p>\n

47.
\n\"Graph<\/p>\n

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.<\/p>\n

48. [latex]\\begin{array}{cc}{a}_{1}=1,& r=\\frac{1}{2}\\end{array}[\/latex]<\/p>\n

49. [latex]\\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n - 1}\\end{array}[\/latex]<\/p>\n

50. [latex]{a}_{n}=27\\cdot {0.3}^{n - 1}[\/latex]<\/p>\n

51. Use recursive formulas to give two examples of geometric sequences whose 3rd<\/sup> terms are [latex]200[\/latex].<\/p>\n

52.\u00a0Use explicit formulas to give two examples of geometric sequences whose 7th<\/sup> terms are [latex]1024[\/latex].<\/p>\n

53. Find the 5th<\/sup> term of the geometric sequence [latex]\\left\\{b,4b,16b,...\\right\\}[\/latex].<\/p>\n

54.\u00a0Find the 7th<\/sup> term of the geometric sequence [latex]\\left\\{64a\\left(-b\\right),32a\\left(-3b\\right),16a\\left(-9b\\right),...\\right\\}[\/latex].<\/p>\n

55. At which term does the sequence [latex]\\left\\{10,12,14.4,17.28,\\text{ }...\\right\\}[\/latex] exceed [latex]100?[\/latex]<\/p>\n

56.\u00a0At which term does the sequence [latex]\\left\\{\\frac{1}{2187},\\frac{1}{729},\\frac{1}{243},\\frac{1}{81}\\text{ }...\\right\\}[\/latex] begin to have integer values?<\/p>\n

57. For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\\left(\\frac{2}{3}\\right)}^{n - 1}[\/latex] first have a non-integer value?<\/p>\n

58.\u00a0Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10th<\/sup> term.<\/p>\n

59. 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<\/sup> term.<\/p>\n

60.\u00a0Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.<\/p>\n\n\t\t\t

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