{"id":15630,"date":"2019-09-05T17:15:42","date_gmt":"2019-09-05T17:15:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15630"},"modified":"2021-06-30T15:45:36","modified_gmt":"2021-06-30T15:45:36","slug":"problem-set-37-sequences-and-their-notations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/problem-set-37-sequences-and-their-notations\/","title":{"raw":"Module 2 Problem Set","rendered":"Module 2 Problem Set"},"content":{"raw":"

Sequences and Their Notation<\/h2>\r\n1. Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?\r\n\r\n2.\u00a0Describe three ways that a sequence can be defined.\r\n\r\n3. Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.\r\n\r\n4.\u00a0What happens to the terms [latex]{a}_{n}[\/latex] of a sequence when there is a negative factor in the formula that is raised to a power that includes [latex]n?[\/latex] What is the term used to describe this phenomenon?\r\n\r\n5. What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.\r\n\r\nFor the following exercises, write the first four terms of the sequence.\r\n\r\n6. [latex]{a}_{n}={2}^{n}-2[\/latex]\r\n\r\n7. [latex]{a}_{n}=-\\frac{16}{n+1}[\/latex]\r\n\r\n8.\u00a0[latex]{a}_{n}=-{\\left(-5\\right)}^{n - 1}[\/latex]\r\n\r\n9. [latex]{a}_{n}=\\frac{{2}^{n}}{{n}^{3}}[\/latex]\r\n\r\n10.\u00a0[latex]{a}_{n}=\\frac{2n+1}{{n}^{3}}[\/latex]\r\n\r\n11. [latex]{a}_{n}=1.25\\cdot {\\left(-4\\right)}^{n - 1}[\/latex]\r\n\r\n12.\u00a0[latex]{a}_{n}=-4\\cdot {\\left(-6\\right)}^{n - 1}[\/latex]\r\n\r\n13. [latex]{a}_{n}=\\frac{{n}^{2}}{2n+1}[\/latex]\r\n\r\n14.\u00a0[latex]{a}_{n}={\\left(-10\\right)}^{n}+1[\/latex]\r\n\r\n15. [latex]{a}_{n}=-\\left(\\frac{4\\cdot {\\left(-5\\right)}^{n - 1}}{5}\\right)[\/latex]\r\n\r\nFor the following exercises, write the first five terms of the sequence.\r\n\r\n16. [latex]{a}_{1}=9,\\text{ }{a}_{n}={a}_{n - 1}+n[\/latex]\r\n\r\n17. [latex]{a}_{1}=3,\\text{ }{a}_{n}=\\left(-3\\right){a}_{n - 1}[\/latex]\r\n\r\n18.\u00a0[latex]{a}_{1}=-4,\\text{ }{a}_{n}=\\frac{{a}_{n - 1}+2n}{{a}_{n - 1}-1}[\/latex]\r\n\r\n19. [latex]{a}_{1}=-1,\\text{ }{a}_{n}=\\frac{{\\left(-3\\right)}^{n - 1}}{{a}_{n - 1}-2}[\/latex]\r\n\r\n20.\u00a0[latex]{a}_{1}=-30,\\text{ }{a}_{n}=\\left(2+{a}_{n - 1}\\right){\\left(\\frac{1}{2}\\right)}^{n}[\/latex]\r\n\r\nFor the following exercises, write the first eight terms of the sequence.\r\n\r\n21. [latex]{a}_{1}=\\frac{1}{24},{\\text{ a}}_{2}=1,\\text{ }{a}_{n}=\\left(2{a}_{n - 2}\\right)\\left(3{a}_{n - 1}\\right)[\/latex]\r\n\r\n22.\u00a0[latex]{a}_{1}=-1,{\\text{ a}}_{2}=5,\\text{ }{a}_{n}={a}_{n - 2}\\left(3-{a}_{n - 1}\\right)[\/latex]\r\n\r\n23. [latex]{a}_{1}=2,{\\text{ a}}_{2}=10,\\text{ }{a}_{n}=\\frac{2\\left({a}_{n - 1}+2\\right)}{{a}_{n - 2}}[\/latex]\r\n\r\nFor the following exercises, write a explicit formula for each sequence.\r\n\r\n24. [latex]-2.5,-5,-10,-20,-40,\\dots [\/latex]\r\n\r\n25. [latex]-8,-6,-3,1,6,\\dots [\/latex]\r\n\r\n26.\u00a0[latex]2,\\text{ }4,\\text{ }12,\\text{ }48,\\text{ }240,\\text{ }\\dots [\/latex]\r\n\r\n27. [latex]35,\\text{ }38,\\text{ }41,\\text{ }44,\\text{ }47,\\text{ }\\dots [\/latex]\r\n\r\n28.\u00a0[latex]15,3,\\frac{3}{5},\\frac{3}{25},\\frac{3}{125},\\cdots [\/latex]\r\n\r\nFor the following exercises, evaluate the factorial.\r\n\r\n29. [latex]6![\/latex]\r\n\r\n30.\u00a0[latex]\\left(\\frac{12}{6}\\right)![\/latex]\r\n\r\n31. [latex]\\frac{12!}{6!}[\/latex]\r\n\r\n32.\u00a0[latex]\\frac{100!}{99!}[\/latex]\r\n\r\nFor the following exercises, write the first four terms of the sequence.\r\n\r\n33. [latex]{a}_{n}=\\frac{n!}{{n}^{\\text{2}}}[\/latex]\r\n\r\n34.\u00a0[latex]{a}_{n}=\\frac{3\\cdot n!}{4\\cdot n!}[\/latex]\r\n\r\n35. [latex]{a}_{n}=\\frac{n!}{{n}^{2}-n - 1}[\/latex]\r\n\r\n36.\u00a0[latex]{a}_{n}=\\frac{100\\cdot n}{n\\left(n - 1\\right)!}[\/latex]\r\n

Arithmetic Sequences<\/h2>\r\n1. What is an arithmetic sequence?\r\n\r\n2.\u00a0How is the common difference of an arithmetic sequence found?\r\n\r\n3. How do we determine whether a sequence is arithmetic?\r\n\r\n4. Give an example of an arithmetic sequence.\r\n\r\n5. Give an example of a sequence that is not arithmetic.\r\n\r\nFor the following exercises, find the common difference for the arithmetic sequence provided.\r\n\r\n6. [latex]\\left\\{5,11,17,23,29,...\\right\\}[\/latex]\r\n\r\n7. [latex]\\left\\{0,\\frac{1}{2},1,\\frac{3}{2},2,...\\right\\}[\/latex]\r\n\r\nFor the following exercises, determine whether the sequence is arithmetic. If so find the common difference.\r\n\r\n8. [latex]\\left\\{11.4,9.3,7.2,5.1,3,...\\right\\}[\/latex]\r\n\r\n9. [latex]\\left\\{4,16,64,256,1024,...\\right\\}[\/latex]\r\n\r\nFor the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.\r\n\r\n10. [latex]{a}_{1}=-25[\/latex] , [latex]d=-9[\/latex]\r\n\r\n11. [latex]{a}_{1}=0[\/latex] , [latex]d=\\frac{2}{3}[\/latex]\r\n\r\nFor the following exercises, write the first five terms of the arithmetic series given two terms.\r\n\r\n12. [latex]{a}_{1}=17,{a}_{7}=-31[\/latex]\r\n\r\n13. [latex]{a}_{13}=-60,{a}_{33}=-160[\/latex]\r\n\r\nFor the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.\r\n\r\n14. First term is 3, common difference is 4, find the 5th<\/sup> term.\r\n\r\n15. First term is 4, common difference is 5, find the 4th<\/sup> term.\r\n\r\n16.\u00a0First term is 5, common difference is 6, find the 8th<\/sup> term.\r\n\r\n17. First term is 6, common difference is 7, find the 6th<\/sup> term.\r\n\r\n18.\u00a0First term is 7, common difference is 8, find the 7th<\/sup> term.\r\n\r\nFor the following exercises, find the first term given two terms from an arithmetic sequence.\r\n\r\n19. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{6}=12[\/latex] and [latex]{a}_{14}=28[\/latex].\r\n\r\n20.\u00a0Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{7}=21[\/latex] and [latex]{a}_{15}=42[\/latex].\r\n\r\n21. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{8}=40[\/latex] and [latex]{a}_{23}=115[\/latex].\r\n\r\n22.\u00a0Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{9}=54[\/latex] and [latex]{a}_{17}=102[\/latex].\r\n\r\n23. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{11}=11[\/latex] and [latex]{a}_{21}=16[\/latex].\r\n\r\nFor the following exercises, find the specified term given two terms from an arithmetic sequence.\r\n\r\n24. [latex]{a}_{1}=33[\/latex] and [latex]{a}_{7}=-15[\/latex]. Find [latex]{a}_{4}[\/latex].\r\n\r\n25. [latex]{a}_{3}=-17.1[\/latex] and [latex]{a}_{10}=-15.7[\/latex]. Find [latex]{a}_{21}[\/latex].\r\n\r\nFor the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.\r\n\r\n26. [latex]{a}_{n}=24 - 4n[\/latex]\r\n\r\n27.\u00a0[latex]{a}_{n}=\\frac{1}{2}n-\\frac{1}{2}[\/latex]\r\n\r\nFor the following exercises, write an explicit formula for each arithmetic sequence.\r\n\r\n28. [latex]{a}_{n}=\\left\\{3,5,7,...\\right\\}[\/latex]\r\n\r\n29.\u00a0[latex]{a}_{n}=\\left\\{32,24,16,...\\right\\}[\/latex]\r\n\r\n30. [latex]{a}_{n}=\\left\\{-5\\text{, }95\\text{, }195\\text{, }...\\right\\}[\/latex]\r\n\r\n31.\u00a0[latex]{a}_{n}=\\left\\{-17\\text{, }-217\\text{, }-417\\text{,}...\\right\\}[\/latex]\r\n\r\n32. [latex]{a}_{n}=\\left\\{1.8\\text{, }3.6\\text{, }5.4\\text{, }...\\right\\}[\/latex]\r\n\r\n33. [latex]{a}_{n}=\\left\\{15.8,18.5,21.2,...\\right\\}[\/latex]<\/span>\r\n\r\n34.\u00a0[latex]{a}_{n}=\\left\\{\\frac{1}{3},-\\frac{4}{3},-3\\text{, }...\\right\\}[\/latex]\r\n\r\n35. [latex]{a}_{n}=\\left\\{0,\\frac{1}{3},\\frac{2}{3},...\\right\\}[\/latex]\r\n\r\n36.\u00a0[latex]{a}_{n}=\\left\\{-5,-\\frac{10}{3},-\\frac{5}{3},\\dots \\right\\}[\/latex]\r\n\r\nFor the following exercises, find the number of terms in the given finite arithmetic sequence.\r\n\r\n37. [latex]{a}_{n}=\\left\\{3\\text{,}-4\\text{,}-11\\text{, }...\\text{,}-60\\right\\}[\/latex]\r\n\r\n38.\u00a0[latex]{a}_{n}=\\left\\{1.2,1.4,1.6,...,3.8\\right\\}[\/latex]\r\n\r\n39. [latex]{a}_{n}=\\left\\{\\frac{1}{2},2,\\frac{7}{2},...,8\\right\\}[\/latex]\r\n

Geometric Sequences<\/h2>\r\n1. 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 an explicit 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, find the number of terms in the given finite geometric sequence.\r\n\r\n34. [latex]{a}_{n}=\\left\\{-1,3,-9,...,2187\\right\\}[\/latex]\r\n\r\n35. [latex]{a}_{n}=\\left\\{2,1,\\frac{1}{2},...,\\frac{1}{1024}\\right\\}[\/latex]\r\n

Fractals<\/h2>\r\nUsing the initiator and generator shown, draw the next two stages of the iterated fractal.\r\n\r\n\r\n\r\n\r\n\r\n
1.\r\n\r\n\"Initiator<\/td>\r\n2.\r\n\r\n\"Initiator<\/td>\r\n<\/tr>\r\n
3.\r\n\r\n\"Initiator<\/td>\r\n4.\r\n\r\n\"Initiator<\/td>\r\n<\/tr>\r\n
5.\r\n\r\n\"Initiator<\/td>\r\n6.\r\n\r\n\"Initiator<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
    \r\n \t
  1. Create your own version of Sierpinski gasket with added randomness.<\/li>\r\n \t
  2. Create a version of the branching tree fractal from example #3 with added randomness.<\/li>\r\n \t
  3. Determine the fractal dimension of the Koch curve.<\/li>\r\n \t
  4. Determine the fractal dimension of the curve generated in exercise #1<\/li>\r\n \t
  5. Determine the fractal dimension of the Sierpinski carpet generated in exercise #5<\/li>\r\n \t
  6. Determine the fractal dimension of the Cantor set generated in exercise #4<\/li>\r\n<\/ol>","rendered":"

    Sequences and Their Notation<\/h2>\n

    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?<\/p>\n

    2.\u00a0Describe three ways that a sequence can be defined.<\/p>\n

    3. Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.<\/p>\n

    4.\u00a0What happens to the terms [latex]{a}_{n}[\/latex] of a sequence when there is a negative factor in the formula that is raised to a power that includes [latex]n?[\/latex] What is the term used to describe this phenomenon?<\/p>\n

    5. What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.<\/p>\n

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

    6. [latex]{a}_{n}={2}^{n}-2[\/latex]<\/p>\n

    7. [latex]{a}_{n}=-\\frac{16}{n+1}[\/latex]<\/p>\n

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

    9. [latex]{a}_{n}=\\frac{{2}^{n}}{{n}^{3}}[\/latex]<\/p>\n

    10.\u00a0[latex]{a}_{n}=\\frac{2n+1}{{n}^{3}}[\/latex]<\/p>\n

    11. [latex]{a}_{n}=1.25\\cdot {\\left(-4\\right)}^{n - 1}[\/latex]<\/p>\n

    12.\u00a0[latex]{a}_{n}=-4\\cdot {\\left(-6\\right)}^{n - 1}[\/latex]<\/p>\n

    13. [latex]{a}_{n}=\\frac{{n}^{2}}{2n+1}[\/latex]<\/p>\n

    14.\u00a0[latex]{a}_{n}={\\left(-10\\right)}^{n}+1[\/latex]<\/p>\n

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

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

    16. [latex]{a}_{1}=9,\\text{ }{a}_{n}={a}_{n - 1}+n[\/latex]<\/p>\n

    17. [latex]{a}_{1}=3,\\text{ }{a}_{n}=\\left(-3\\right){a}_{n - 1}[\/latex]<\/p>\n

    18.\u00a0[latex]{a}_{1}=-4,\\text{ }{a}_{n}=\\frac{{a}_{n - 1}+2n}{{a}_{n - 1}-1}[\/latex]<\/p>\n

    19. [latex]{a}_{1}=-1,\\text{ }{a}_{n}=\\frac{{\\left(-3\\right)}^{n - 1}}{{a}_{n - 1}-2}[\/latex]<\/p>\n

    20.\u00a0[latex]{a}_{1}=-30,\\text{ }{a}_{n}=\\left(2+{a}_{n - 1}\\right){\\left(\\frac{1}{2}\\right)}^{n}[\/latex]<\/p>\n

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

    21. [latex]{a}_{1}=\\frac{1}{24},{\\text{ a}}_{2}=1,\\text{ }{a}_{n}=\\left(2{a}_{n - 2}\\right)\\left(3{a}_{n - 1}\\right)[\/latex]<\/p>\n

    22.\u00a0[latex]{a}_{1}=-1,{\\text{ a}}_{2}=5,\\text{ }{a}_{n}={a}_{n - 2}\\left(3-{a}_{n - 1}\\right)[\/latex]<\/p>\n

    23. [latex]{a}_{1}=2,{\\text{ a}}_{2}=10,\\text{ }{a}_{n}=\\frac{2\\left({a}_{n - 1}+2\\right)}{{a}_{n - 2}}[\/latex]<\/p>\n

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

    24. [latex]-2.5,-5,-10,-20,-40,\\dots[\/latex]<\/p>\n

    25. [latex]-8,-6,-3,1,6,\\dots[\/latex]<\/p>\n

    26.\u00a0[latex]2,\\text{ }4,\\text{ }12,\\text{ }48,\\text{ }240,\\text{ }\\dots[\/latex]<\/p>\n

    27. [latex]35,\\text{ }38,\\text{ }41,\\text{ }44,\\text{ }47,\\text{ }\\dots[\/latex]<\/p>\n

    28.\u00a0[latex]15,3,\\frac{3}{5},\\frac{3}{25},\\frac{3}{125},\\cdots[\/latex]<\/p>\n

    For the following exercises, evaluate the factorial.<\/p>\n

    29. [latex]6![\/latex]<\/p>\n

    30.\u00a0[latex]\\left(\\frac{12}{6}\\right)![\/latex]<\/p>\n

    31. [latex]\\frac{12!}{6!}[\/latex]<\/p>\n

    32.\u00a0[latex]\\frac{100!}{99!}[\/latex]<\/p>\n

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

    33. [latex]{a}_{n}=\\frac{n!}{{n}^{\\text{2}}}[\/latex]<\/p>\n

    34.\u00a0[latex]{a}_{n}=\\frac{3\\cdot n!}{4\\cdot n!}[\/latex]<\/p>\n

    35. [latex]{a}_{n}=\\frac{n!}{{n}^{2}-n - 1}[\/latex]<\/p>\n

    36.\u00a0[latex]{a}_{n}=\\frac{100\\cdot n}{n\\left(n - 1\\right)!}[\/latex]<\/p>\n

    Arithmetic Sequences<\/h2>\n

    1. What is an arithmetic sequence?<\/p>\n

    2.\u00a0How is the common difference of an arithmetic sequence found?<\/p>\n

    3. How do we determine whether a sequence is arithmetic?<\/p>\n

    4. Give an example of an arithmetic sequence.<\/p>\n

    5. Give an example of a sequence that is not arithmetic.<\/p>\n

    For the following exercises, find the common difference for the arithmetic sequence provided.<\/p>\n

    6. [latex]\\left\\{5,11,17,23,29,...\\right\\}[\/latex]<\/p>\n

    7. [latex]\\left\\{0,\\frac{1}{2},1,\\frac{3}{2},2,...\\right\\}[\/latex]<\/p>\n

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

    8. [latex]\\left\\{11.4,9.3,7.2,5.1,3,...\\right\\}[\/latex]<\/p>\n

    9. [latex]\\left\\{4,16,64,256,1024,...\\right\\}[\/latex]<\/p>\n

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

    10. [latex]{a}_{1}=-25[\/latex] , [latex]d=-9[\/latex]<\/p>\n

    11. [latex]{a}_{1}=0[\/latex] , [latex]d=\\frac{2}{3}[\/latex]<\/p>\n

    For the following exercises, write the first five terms of the arithmetic series given two terms.<\/p>\n

    12. [latex]{a}_{1}=17,{a}_{7}=-31[\/latex]<\/p>\n

    13. [latex]{a}_{13}=-60,{a}_{33}=-160[\/latex]<\/p>\n

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

    14. First term is 3, common difference is 4, find the 5th<\/sup> term.<\/p>\n

    15. First term is 4, common difference is 5, find the 4th<\/sup> term.<\/p>\n

    16.\u00a0First term is 5, common difference is 6, find the 8th<\/sup> term.<\/p>\n

    17. First term is 6, common difference is 7, find the 6th<\/sup> term.<\/p>\n

    18.\u00a0First term is 7, common difference is 8, find the 7th<\/sup> term.<\/p>\n

    For the following exercises, find the first term given two terms from an arithmetic sequence.<\/p>\n

    19. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{6}=12[\/latex] and [latex]{a}_{14}=28[\/latex].<\/p>\n

    20.\u00a0Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{7}=21[\/latex] and [latex]{a}_{15}=42[\/latex].<\/p>\n

    21. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{8}=40[\/latex] and [latex]{a}_{23}=115[\/latex].<\/p>\n

    22.\u00a0Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{9}=54[\/latex] and [latex]{a}_{17}=102[\/latex].<\/p>\n

    23. Find the first term or [latex]{a}_{1}[\/latex] of an arithmetic sequence if [latex]{a}_{11}=11[\/latex] and [latex]{a}_{21}=16[\/latex].<\/p>\n

    For the following exercises, find the specified term given two terms from an arithmetic sequence.<\/p>\n

    24. [latex]{a}_{1}=33[\/latex] and [latex]{a}_{7}=-15[\/latex]. Find [latex]{a}_{4}[\/latex].<\/p>\n

    25. [latex]{a}_{3}=-17.1[\/latex] and [latex]{a}_{10}=-15.7[\/latex]. Find [latex]{a}_{21}[\/latex].<\/p>\n

    For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.<\/p>\n

    26. [latex]{a}_{n}=24 - 4n[\/latex]<\/p>\n

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

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

    28. [latex]{a}_{n}=\\left\\{3,5,7,...\\right\\}[\/latex]<\/p>\n

    29.\u00a0[latex]{a}_{n}=\\left\\{32,24,16,...\\right\\}[\/latex]<\/p>\n

    30. [latex]{a}_{n}=\\left\\{-5\\text{, }95\\text{, }195\\text{, }...\\right\\}[\/latex]<\/p>\n

    31.\u00a0[latex]{a}_{n}=\\left\\{-17\\text{, }-217\\text{, }-417\\text{,}...\\right\\}[\/latex]<\/p>\n

    32. [latex]{a}_{n}=\\left\\{1.8\\text{, }3.6\\text{, }5.4\\text{, }...\\right\\}[\/latex]<\/p>\n

    33. [latex]{a}_{n}=\\left\\{15.8,18.5,21.2,...\\right\\}[\/latex]<\/span><\/p>\n

    34.\u00a0[latex]{a}_{n}=\\left\\{\\frac{1}{3},-\\frac{4}{3},-3\\text{, }...\\right\\}[\/latex]<\/p>\n

    35. [latex]{a}_{n}=\\left\\{0,\\frac{1}{3},\\frac{2}{3},...\\right\\}[\/latex]<\/p>\n

    36.\u00a0[latex]{a}_{n}=\\left\\{-5,-\\frac{10}{3},-\\frac{5}{3},\\dots \\right\\}[\/latex]<\/p>\n

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

    37. [latex]{a}_{n}=\\left\\{3\\text{,}-4\\text{,}-11\\text{, }...\\text{,}-60\\right\\}[\/latex]<\/p>\n

    38.\u00a0[latex]{a}_{n}=\\left\\{1.2,1.4,1.6,...,3.8\\right\\}[\/latex]<\/p>\n

    39. [latex]{a}_{n}=\\left\\{\\frac{1}{2},2,\\frac{7}{2},...,8\\right\\}[\/latex]<\/p>\n

    Geometric Sequences<\/h2>\n

    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 an explicit 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, find the number of terms in the given finite geometric sequence.<\/p>\n

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

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

    Fractals<\/h2>\n

    Using the initiator and generator shown, draw the next two stages of the iterated fractal.<\/p>\n\n\n\n\n\n
    1.<\/p>\n

    \"Initiator<\/td>\n

    		2.<\/p>\n

    \"Initiator<\/td>\n<\/tr>\n

    3.<\/p>\n

    \"Initiator<\/td>\n

    		4.<\/p>\n

    \"Initiator<\/td>\n<\/tr>\n

    5.<\/p>\n

    \"Initiator<\/td>\n

    		6.<\/p>\n

    \"Initiator<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \n
    1. Create your own version of Sierpinski gasket with added randomness.<\/li>\n
    2. Create a version of the branching tree fractal from example #3 with added randomness.<\/li>\n
    3. Determine the fractal dimension of the Koch curve.<\/li>\n
    4. Determine the fractal dimension of the curve generated in exercise #1<\/li>\n
    5. Determine the fractal dimension of the Sierpinski carpet generated in exercise #5<\/li>\n
    6. Determine the fractal dimension of the Cantor set generated in exercise #4<\/li>\n<\/ol>\n\n\t\t\t
      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
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