Section Exercises

1. What is a geometric sequence? 2. How is the common ratio of a geometric sequence found? 3. What is the procedure for determining whether a sequence is geometric? 4. What is the difference between an arithmetic sequence and a geometric sequence? 5. 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. [latex]1,3,9,27,81,..[/latex]. 7. [latex]-0.125,0.25,-0.5,1,-2,..[/latex]. 8. [latex]-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128},..[/latex]. For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 9. [latex]-6,-12,-24,-48,-96,..[/latex]. 10. [latex]5,5.2,5.4,5.6,5.8,..[/latex]. 11. [latex]-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},..[/latex]. 12. [latex]6,8,11,15,20,..[/latex]. 13. [latex]0.8,4,20,100,500,..[/latex]. For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. 14. [latex]\begin{array}{cc}{a}_{1}=8,& r=0.3\end{array}[/latex] 15. [latex]\begin{array}{cc}{a}_{1}=5,& r=\frac{1}{5}\end{array}[/latex] For the following exercises, write the first five terms of the geometric sequence, given any two terms. 16. [latex]\begin{array}{cc}{a}_{7}=64,& {a}_{10}\end{array}=512[/latex] 17. [latex]\begin{array}{cc}{a}_{6}=25,& {a}_{8}\end{array}=6.25[/latex] For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. 18. The first term is [latex]2[/latex], and the common ratio is [latex]3[/latex]. Find the 5^th term. 19. The first term is 16 and the common ratio is [latex]-\frac{1}{3}[/latex]. Find the 4^th term. For the following exercises, find the specified term for the geometric sequence, given the first four terms. 20. [latex]{a}_{n}=\left\{-1,2,-4,8,...\right\}[/latex]. Find [latex]{a}_{12}[/latex]. 21. [latex]{a}_{n}=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},...\right\}[/latex]. Find [latex]{a}_{7}[/latex]. For the following exercises, write the first five terms of the geometric sequence. 22. [latex]\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\frac{1}{3}\end{array}{a}_{n - 1}[/latex] 23. [latex]\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\end{array}[/latex] For the following exercises, write a recursive formula for each geometric sequence. 24. [latex]{a}_{n}=\left\{-1,5,-25,125,...\right\}[/latex] 25. [latex]{a}_{n}=\left\{-32,-16,-8,-4,...\right\}[/latex] 26. [latex]{a}_{n}=\left\{14,56,224,896,...\right\}[/latex] 27. [latex]{a}_{n}=\left\{10,-3,0.9,-0.27,...\right\}[/latex] 28. [latex]{a}_{n}=\left\{0.61,1.83,5.49,16.47,...\right\}[/latex] 29. [latex]{a}_{n}=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},...\right\}[/latex] 30. [latex]{a}_{n}=\left\{-2,\frac{4}{3},-\frac{8}{9},\frac{16}{27},...\right\}[/latex] 31. [latex]{a}_{n}=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},...\right\}[/latex] For the following exercises, write the first five terms of the geometric sequence. 32. [latex]{a}_{n}=-4\cdot {5}^{n - 1}[/latex] 33. [latex]{a}_{n}=12\cdot {\left(-\frac{1}{2}\right)}^{n - 1}[/latex] For the following exercises, write an explicit formula for each geometric sequence. 34. [latex]{a}_{n}=\left\{-2,-4,-8,-16,...\right\}[/latex] 35. [latex]{a}_{n}=\left\{1,3,9,27,...\right\}[/latex] 36. [latex]{a}_{n}=\left\{-4,-12,-36,-108,...\right\}[/latex] 37. [latex]{a}_{n}=\left\{0.8,-4,20,-100,...\right\}[/latex] 38. [latex]{a}_{n}=\left\{-1.25,-5,-20,-80,...\right\}[/latex] 39. [latex]{a}_{n}=\left\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125},...\right\}[/latex] 40. [latex]{a}_{n}=\left\{2,\frac{1}{3},\frac{1}{18},\frac{1}{108},...\right\}[/latex] 41. [latex]{a}_{n}=\left\{3,-1,\frac{1}{3},-\frac{1}{9},...\right\}[/latex] For the following exercises, find the specified term for the geometric sequence given. 42. Let [latex]{a}_{1}=4[/latex], [latex]{a}_{n}=-3{a}_{n - 1}[/latex]. Find [latex]{a}_{8}[/latex]. 43. Let [latex]{a}_{n}=-{\left(-\frac{1}{3}\right)}^{n - 1}[/latex]. Find [latex]{a}_{12}[/latex]. For the following exercises, find the number of terms in the given finite geometric sequence. 44. [latex]{a}_{n}=\left\{-1,3,-9,...,2187\right\}[/latex] 45. [latex]{a}_{n}=\left\{2,1,\frac{1}{2},...,\frac{1}{1024}\right\}[/latex] For the following exercises, determine whether the graph shown represents a geometric sequence. 46. 47. For the following exercises, use the information provided to graph the first five terms of the geometric sequence. 48. [latex]\begin{array}{cc}{a}_{1}=1,& r=\frac{1}{2}\end{array}[/latex] 49. [latex]\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n - 1}\end{array}[/latex] 50. [latex]{a}_{n}=27\cdot {0.3}^{n - 1}[/latex] 51. Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are [latex]200[/latex]. 52. Use explicit formulas to give two examples of geometric sequences whose 7^th terms are [latex]1024[/latex]. 53. Find the 5^th term of the geometric sequence [latex]\left\{b,4b,16b,...\right\}[/latex]. 54. Find the 7^th term of the geometric sequence [latex]\left\{64a\left(-b\right),32a\left(-3b\right),16a\left(-9b\right),...\right\}[/latex]. 55. At which term does the sequence [latex]\left\{10,12,14.4,17.28,\text{ }...\right\}[/latex] exceed [latex]100?[/latex] 56. At 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? 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? 58. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10^th term. 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 8^th term. 60. Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

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