{"id":2064,"date":"2015-11-12T18:30:42","date_gmt":"2015-11-12T18:30:42","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2064"},"modified":"2015-11-12T18:30:42","modified_gmt":"2015-11-12T18:30:42","slug":"solutions-12","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solutions-12\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\n1.\u00a0The sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex] .\n\n2.\u00a0The sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex] .\n\n3.\u00a0[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]\n\n4.\u00a0[latex]\\begin{array}{l}{a}_{1}=2\\\\ {a}_{n}=\\frac{2}{3}{a}_{n - 1}\\text{ for }n\\ge 2\\end{array}[\/latex]\n\n5.\u00a0[latex]{a}_{6}=16,384[\/latex]\n\n6.\u00a0[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]\n\n7. a.\u00a0[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]\nb. The number of hits will be about 333.\n\n

Solutions to Odd-Numbered Exercises<\/h2>\n1.\u00a0A sequence in which the ratio between any two consecutive terms is constant.\n\n3.\u00a0Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.\n\n5.\u00a0Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.\n\n7.\u00a0The common ratio is [latex]-2[\/latex]\n\n9.\u00a0The sequence is geometric. The common ratio is 2.\n\n11.\u00a0The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}[\/latex].\n\n13.\u00a0The sequence is geometric. The common ratio is [latex]5[\/latex].\n\n15.\u00a0[latex]5,1,\\frac{1}{5},\\frac{1}{25},\\frac{1}{125}[\/latex]\n\n17.\u00a0[latex]800,400,200,100,50[\/latex]\n\n19.\u00a0[latex]{a}_{4}=-\\frac{16}{27}[\/latex]\n\n21.\u00a0[latex]{a}_{7}=-\\frac{2}{729}[\/latex]\n\n23.\u00a0[latex]7,1.4,0.28,0.056,0.0112[\/latex]\n\n25.\u00a0[latex]\\begin{array}{cc}a{}_{1}=-32,& {a}_{n}=\\frac{1}{2}{a}_{n - 1}\\end{array}[\/latex]\n\n27.\u00a0[latex]\\begin{array}{cc}{a}_{1}=10,& {a}_{n}=-0.3{a}_{n - 1}\\end{array}[\/latex]\n\n29.\u00a0[latex]\\begin{array}{cc}{a}_{1}=\\frac{3}{5},& {a}_{n}=\\frac{1}{6}{a}_{n - 1}\\end{array}[\/latex]\n\n31.\u00a0[latex]{a}_{1}=\\frac{1}{512},{a}_{n}=-4{a}_{n - 1}[\/latex]\n\n33.\u00a0[latex]12,-6,3,-\\frac{3}{2},\\frac{3}{4}[\/latex]\n\n35.\u00a0[latex]{a}_{n}={3}^{n - 1}[\/latex]\n\n37.\u00a0[latex]{a}_{n}=0.8\\cdot {\\left(-5\\right)}^{n - 1}[\/latex]\n\n39.\u00a0[latex]{a}_{n}=-{\\left(\\frac{4}{5}\\right)}^{n - 1}[\/latex]\n\n41.\u00a0[latex]{a}_{n}=3\\cdot {\\left(-\\frac{1}{3}\\right)}^{n - 1}[\/latex]\n\n43.\u00a0[latex]{a}_{12}=\\frac{1}{177,147}[\/latex]\n\n45.\u00a0There are [latex]12[\/latex] terms in the sequence.\n\n47.\u00a0The graph does not represent a geometric sequence.\n\n49.\n\"Graph\n\n51.\u00a0Answers will vary. Examples: [latex]{\\begin{array}{cc}{a}_{1}=800,& {a}_{n}=0.5a\\end{array}}_{n - 1}[\/latex] and [latex]{\\begin{array}{cc}{a}_{1}=12.5,& {a}_{n}=4a\\end{array}}_{n - 1}[\/latex]\n\n53.\u00a0[latex]{a}_{5}=256b[\/latex]\n\n55.\u00a0The sequence exceeds [latex]100[\/latex] at the 14th<\/sup> term, [latex]{a}_{14}\\approx 107[\/latex].\n\n57.\u00a0[latex]{a}_{4}=-\\frac{32}{3}[\/latex] is the first non-integer value\n\n59.\u00a0Answers will vary. Example: Explicit formula with a decimal common ratio: [latex]{a}_{n}=400\\cdot {0.5}^{n - 1}[\/latex]; First 4 terms: [latex]\\begin{array}{cc}400,200,100,50;& {a}_{8}=3.125\\end{array}[\/latex]\n","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0The sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex] .<\/p>\n

2.\u00a0The sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex] .<\/p>\n

3.\u00a0[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]<\/p>\n

4.\u00a0[latex]\\begin{array}{l}{a}_{1}=2\\\\ {a}_{n}=\\frac{2}{3}{a}_{n - 1}\\text{ for }n\\ge 2\\end{array}[\/latex]<\/p>\n

5.\u00a0[latex]{a}_{6}=16,384[\/latex]<\/p>\n

6.\u00a0[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]<\/p>\n

7. a.\u00a0[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]
\nb. The number of hits will be about 333.<\/p>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0A sequence in which the ratio between any two consecutive terms is constant.<\/p>\n

3.\u00a0Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.<\/p>\n

5.\u00a0Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.<\/p>\n

7.\u00a0The common ratio is [latex]-2[\/latex]<\/p>\n

9.\u00a0The sequence is geometric. The common ratio is 2.<\/p>\n

11.\u00a0The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}[\/latex].<\/p>\n

13.\u00a0The sequence is geometric. The common ratio is [latex]5[\/latex].<\/p>\n

15.\u00a0[latex]5,1,\\frac{1}{5},\\frac{1}{25},\\frac{1}{125}[\/latex]<\/p>\n

17.\u00a0[latex]800,400,200,100,50[\/latex]<\/p>\n

19.\u00a0[latex]{a}_{4}=-\\frac{16}{27}[\/latex]<\/p>\n

21.\u00a0[latex]{a}_{7}=-\\frac{2}{729}[\/latex]<\/p>\n

23.\u00a0[latex]7,1.4,0.28,0.056,0.0112[\/latex]<\/p>\n

25.\u00a0[latex]\\begin{array}{cc}a{}_{1}=-32,& {a}_{n}=\\frac{1}{2}{a}_{n - 1}\\end{array}[\/latex]<\/p>\n

27.\u00a0[latex]\\begin{array}{cc}{a}_{1}=10,& {a}_{n}=-0.3{a}_{n - 1}\\end{array}[\/latex]<\/p>\n

29.\u00a0[latex]\\begin{array}{cc}{a}_{1}=\\frac{3}{5},& {a}_{n}=\\frac{1}{6}{a}_{n - 1}\\end{array}[\/latex]<\/p>\n

31.\u00a0[latex]{a}_{1}=\\frac{1}{512},{a}_{n}=-4{a}_{n - 1}[\/latex]<\/p>\n

33.\u00a0[latex]12,-6,3,-\\frac{3}{2},\\frac{3}{4}[\/latex]<\/p>\n

35.\u00a0[latex]{a}_{n}={3}^{n - 1}[\/latex]<\/p>\n

37.\u00a0[latex]{a}_{n}=0.8\\cdot {\\left(-5\\right)}^{n - 1}[\/latex]<\/p>\n

39.\u00a0[latex]{a}_{n}=-{\\left(\\frac{4}{5}\\right)}^{n - 1}[\/latex]<\/p>\n

41.\u00a0[latex]{a}_{n}=3\\cdot {\\left(-\\frac{1}{3}\\right)}^{n - 1}[\/latex]<\/p>\n

43.\u00a0[latex]{a}_{12}=\\frac{1}{177,147}[\/latex]<\/p>\n

45.\u00a0There are [latex]12[\/latex] terms in the sequence.<\/p>\n

47.\u00a0The graph does not represent a geometric sequence.<\/p>\n

49.
\n\"Graph<\/p>\n

51.\u00a0Answers will vary. Examples: [latex]{\\begin{array}{cc}{a}_{1}=800,& {a}_{n}=0.5a\\end{array}}_{n - 1}[\/latex] and [latex]{\\begin{array}{cc}{a}_{1}=12.5,& {a}_{n}=4a\\end{array}}_{n - 1}[\/latex]<\/p>\n

53.\u00a0[latex]{a}_{5}=256b[\/latex]<\/p>\n

55.\u00a0The sequence exceeds [latex]100[\/latex] at the 14th<\/sup> term, [latex]{a}_{14}\\approx 107[\/latex].<\/p>\n

57.\u00a0[latex]{a}_{4}=-\\frac{32}{3}[\/latex] is the first non-integer value<\/p>\n

59.\u00a0Answers will vary. Example: Explicit formula with a decimal common ratio: [latex]{a}_{n}=400\\cdot {0.5}^{n - 1}[\/latex]; First 4 terms: [latex]\\begin{array}{cc}400,200,100,50;& {a}_{8}=3.125\\end{array}[\/latex]<\/p>\n\n\t\t\t

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