{"id":2048,"date":"2015-11-12T18:30:42","date_gmt":"2015-11-12T18:30:42","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2048"},"modified":"2015-11-12T18:30:42","modified_gmt":"2015-11-12T18:30:42","slug":"solutions-14","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/solutions-14\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\n1.\u00a0The sequence is arithmetic. The common difference is [latex]-2[\/latex].\n\n2.\u00a0The sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].\n\n3.\u00a0[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]\n\n4.\u00a0[latex]{a}_{2}=2[\/latex]\n\n5.\u00a0[latex]\\begin{array}{l}{a}_{1}=25\\hfill \\\\ {a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2\\hfill \\end{array}[\/latex]\n\n6.\u00a0[latex]{a}_{n}=53 - 3n[\/latex]\n\n7.\u00a0There are 11 terms in the sequence.\n\n8.\u00a0The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.\n

Solutions to Odd-Numbered Exercises<\/h2>\n1.\u00a0A sequence where each successive term of the sequence increases (or decreases) by a constant value.\n\n3.\u00a0We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.\n\n5.\u00a0Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.\n\n7.\u00a0The common difference is [latex]\\frac{1}{2}[\/latex]\n\n9.\u00a0The sequence is not arithmetic because [latex]16 - 4\\ne 64 - 16[\/latex].\n\n11.\u00a0[latex]0,\\frac{2}{3},\\frac{4}{3},2,\\frac{8}{3}[\/latex]\n\n13.\u00a0[latex]0,-5,-10,-15,-20[\/latex]\n\n15.\u00a0[latex]{a}_{4}=19[\/latex]\n\n17.\u00a0[latex]{a}_{6}=41[\/latex]\n\n19.\u00a0[latex]{a}_{1}=2[\/latex]\n\n21.\u00a0[latex]{a}_{1}=5[\/latex]\n\n23.\u00a0[latex]{a}_{1}=6[\/latex]\n\n25.\u00a0[latex]{a}_{21}=-13.5[\/latex]\n\n27. [latex]-19,-20.4,-21.8,-23.2,-24.6[\/latex]\n\n29. [latex]\\begin{array}{ll}{a}_{1}=17; {a}_{n}={a}_{n - 1}+9\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]\n\n31.\u00a0[latex]\\begin{array}{ll}{a}_{1}=12; {a}_{n}={a}_{n - 1}+5\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]\n\n33.\u00a0[latex]\\begin{array}{ll}{a}_{1}=8.9; {a}_{n}={a}_{n - 1}+1.4\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]\n\n35.\u00a0[latex]\\begin{array}{ll}{a}_{1}=\\frac{1}{5}; {a}_{n}={a}_{n - 1}+\\frac{1}{4}\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]\n\n37.\u00a0[latex]\\begin{array}{ll}{}_{1}=\\frac{1}{6}; {a}_{n}={a}_{n - 1}-\\frac{13}{12}\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]\n\n39.\u00a0[latex]{a}_{1}=4;\\text{ }{a}_{n}={a}_{n - 1}+7;\\text{ }{a}_{14}=95[\/latex]\n\n41.\u00a0First five terms: [latex]20,16,12,8,4[\/latex].\n\n43.\u00a0[latex]{a}_{n}=1+2n[\/latex]\n\n45.\u00a0[latex]{a}_{n}=-105+100n[\/latex]\n\n47.\u00a0[latex]{a}_{n}=1.8n[\/latex]\n\n49.\u00a0[latex]{a}_{n}=13.1+2.7n[\/latex]\n\n51.\u00a0[latex]{a}_{n}=\\frac{1}{3}n-\\frac{1}{3}[\/latex]\n\n53.\u00a0There are 10 terms in the sequence.\n\n55.\u00a0There are 6 terms in the sequence.\n\n57.\u00a0The graph does not represent an arithmetic sequence.\n\n59.\n\"Graph\n\n61.\u00a0[latex]1,4,7,10,13,16,19[\/latex]\n\n63.\n\"Graph\n\n65.\n\"Graph\n\n67.\u00a0Answers will vary. Examples: [latex]{a}_{n}=20.6n[\/latex] and [latex]{a}_{n}=2+20.4\\mathrm{n.}[\/latex]\n\n69.\u00a0[latex]{a}_{11}=-17a+38b[\/latex]\n\n71.\u00a0The sequence begins to have negative values at the 13th<\/sup> term, [latex]{a}_{13}=-\\frac{1}{3}[\/latex]\n\n73.\u00a0Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: [latex]{a}_{1}=3,{a}_{n}={a}_{n - 1}-3[\/latex]. First 4 terms: [latex]\\begin{array}{ll}3,0,-3,-6\\hfill & {a}_{31}=-87\\hfill \\end{array}[\/latex]","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0The sequence is arithmetic. The common difference is [latex]-2[\/latex].<\/p>\n

2.\u00a0The sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].<\/p>\n

3.\u00a0[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/p>\n

4.\u00a0[latex]{a}_{2}=2[\/latex]<\/p>\n

5.\u00a0[latex]\\begin{array}{l}{a}_{1}=25\\hfill \\\\ {a}_{n}={a}_{n - 1}+12,\\text{ for }n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

6.\u00a0[latex]{a}_{n}=53 - 3n[\/latex]<\/p>\n

7.\u00a0There are 11 terms in the sequence.<\/p>\n

8.\u00a0The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.<\/p>\n

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

1.\u00a0A sequence where each successive term of the sequence increases (or decreases) by a constant value.<\/p>\n

3.\u00a0We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.<\/p>\n

5.\u00a0Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.<\/p>\n

7.\u00a0The common difference is [latex]\\frac{1}{2}[\/latex]<\/p>\n

9.\u00a0The sequence is not arithmetic because [latex]16 - 4\\ne 64 - 16[\/latex].<\/p>\n

11.\u00a0[latex]0,\\frac{2}{3},\\frac{4}{3},2,\\frac{8}{3}[\/latex]<\/p>\n

13.\u00a0[latex]0,-5,-10,-15,-20[\/latex]<\/p>\n

15.\u00a0[latex]{a}_{4}=19[\/latex]<\/p>\n

17.\u00a0[latex]{a}_{6}=41[\/latex]<\/p>\n

19.\u00a0[latex]{a}_{1}=2[\/latex]<\/p>\n

21.\u00a0[latex]{a}_{1}=5[\/latex]<\/p>\n

23.\u00a0[latex]{a}_{1}=6[\/latex]<\/p>\n

25.\u00a0[latex]{a}_{21}=-13.5[\/latex]<\/p>\n

27. [latex]-19,-20.4,-21.8,-23.2,-24.6[\/latex]<\/p>\n

29. [latex]\\begin{array}{ll}{a}_{1}=17; {a}_{n}={a}_{n - 1}+9\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

31.\u00a0[latex]\\begin{array}{ll}{a}_{1}=12; {a}_{n}={a}_{n - 1}+5\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

33.\u00a0[latex]\\begin{array}{ll}{a}_{1}=8.9; {a}_{n}={a}_{n - 1}+1.4\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

35.\u00a0[latex]\\begin{array}{ll}{a}_{1}=\\frac{1}{5}; {a}_{n}={a}_{n - 1}+\\frac{1}{4}\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

37.\u00a0[latex]\\begin{array}{ll}{}_{1}=\\frac{1}{6}; {a}_{n}={a}_{n - 1}-\\frac{13}{12}\\hfill & n\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n

39.\u00a0[latex]{a}_{1}=4;\\text{ }{a}_{n}={a}_{n - 1}+7;\\text{ }{a}_{14}=95[\/latex]<\/p>\n

41.\u00a0First five terms: [latex]20,16,12,8,4[\/latex].<\/p>\n

43.\u00a0[latex]{a}_{n}=1+2n[\/latex]<\/p>\n

45.\u00a0[latex]{a}_{n}=-105+100n[\/latex]<\/p>\n

47.\u00a0[latex]{a}_{n}=1.8n[\/latex]<\/p>\n

49.\u00a0[latex]{a}_{n}=13.1+2.7n[\/latex]<\/p>\n

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

53.\u00a0There are 10 terms in the sequence.<\/p>\n

55.\u00a0There are 6 terms in the sequence.<\/p>\n

57.\u00a0The graph does not represent an arithmetic sequence.<\/p>\n

59.
\n\"Graph<\/p>\n

61.\u00a0[latex]1,4,7,10,13,16,19[\/latex]<\/p>\n

63.
\n\"Graph<\/p>\n

65.
\n\"Graph<\/p>\n

67.\u00a0Answers will vary. Examples: [latex]{a}_{n}=20.6n[\/latex] and [latex]{a}_{n}=2+20.4\\mathrm{n.}[\/latex]<\/p>\n

69.\u00a0[latex]{a}_{11}=-17a+38b[\/latex]<\/p>\n

71.\u00a0The sequence begins to have negative values at the 13th<\/sup> term, [latex]{a}_{13}=-\\frac{1}{3}[\/latex]<\/p>\n

73.\u00a0Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: [latex]{a}_{1}=3,{a}_{n}={a}_{n - 1}-3[\/latex]. First 4 terms: [latex]\\begin{array}{ll}3,0,-3,-6\\hfill & {a}_{31}=-87\\hfill \\end{array}[\/latex]<\/p>\n\n\t\t\t

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