{"id":137,"date":"2026-01-12T15:50:25","date_gmt":"2026-01-12T15:50:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=137"},"modified":"2026-01-12T15:50:26","modified_gmt":"2026-01-12T15:50:26","slug":"sequences-and-series-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/sequences-and-series-get-stronger-answer-key\/","title":{"raw":"Sequences and Series: Get Stronger Answer Key","rendered":"Sequences and Series: Get Stronger Answer Key"},"content":{"raw":"

Sequences and Their Notations<\/h2>\r\n
    \r\n \t
  1. First four terms: [latex]-8, -\\frac{16}{5}, -4, -\\frac{16}{5}[\/latex]<\/li>\r\n \t
  2. First four terms: [latex]2, \\frac{1}{2}, \\frac{8}{27}, \\frac{1}{4}[\/latex]<\/li>\r\n \t
  3. First four terms: [latex]1.25, -5, 20, -80[\/latex]<\/li>\r\n \t
  4. First four terms: [latex]\\frac{1}{3}, \\frac{4}{5}, \\frac{9}{7}, \\frac{16}{9}[\/latex]<\/li>\r\n \t
  5. First four terms: [latex]-\\frac{4}{5}, 4, -20, 100[\/latex]<\/li>\r\n \t
  6. [latex]\\frac{1}{3}, \\frac{4}{5}, \\frac{9}{7}, \\frac{16}{9}, \\frac{25}{11}, 31, 44, 59[\/latex]<\/li>\r\n \t
  7. [latex]-0.6, -15, -20, -375, -80, -9375, -320[\/latex]<\/li>\r\n \t
  8. [latex]a_n = n^2 + 3[\/latex]<\/li>\r\n \t
  9. [latex]a_n = \\frac{2^n}{2n}[\/latex] or [latex]2^{n-1}\/n[\/latex]<\/li>\r\n \t
  10. [latex]a_n = (-\\frac{1}{2})^{n-1}[\/latex]<\/li>\r\n \t
  11. First five terms: [latex]3, -9, 27, -81, 243[\/latex]<\/li>\r\n \t
  12. First five terms: [latex]-1, 1, -9, \\frac{27}{11}, \\frac{891}{5}[\/latex]<\/li>\r\n \t
  13. [latex]\\frac{1}{24}, 1, \\frac{3}{4}, \\frac{9}{2}, \\frac{81}{4}, \\frac{531,441}{8}, \\frac{16}{16}[\/latex]<\/li>\r\n \t
  14. [latex]2, 10, 12, \\frac{14}{5}, \\frac{5}{5}, 2, 10, 12[\/latex]<\/li>\r\n \t
  15. [latex]a_1 = -8, a_n = a_{n-1} + n[\/latex]<\/li>\r\n \t
  16. [latex]a_1 = 35, a_n = a_{n-1} + 3[\/latex]<\/li>\r\n \t
  17. [latex]720[\/latex]<\/li>\r\n \t
  18. [latex]665,280[\/latex]<\/li>\r\n \t
  19. First four terms: [latex]1, \\frac{1}{2}, \\frac{2}{3}, \\frac{3}{2}[\/latex]<\/li>\r\n \t
  20. First four terms: [latex]-1, 2, \\frac{5}{6}, \\frac{24}{11}[\/latex]<\/li>\r\n \t
  21. \"Graph<\/li>\r\n \t
  22. \"Graph<\/li>\r\n \t
  23. \"Graph<\/li>\r\n \t
  24. [latex]a_n = 2^{n-2}[\/latex]<\/li>\r\n \t
  25. [latex]a_1 = 6, a_n = 2a_{n-1} - 5[\/latex]<\/li>\r\n<\/ol>\r\n

    Arithmetic Sequences<\/h2>\r\n
      \r\n \t
    1. The common difference is [latex]\\frac{1}{3}[\/latex]<\/li>\r\n \t
    2. The sequence is not arithmetic because [latex]16 - 4 \\neq 64 - 16[\/latex]<\/li>\r\n \t
    3. [latex]0, \\frac{2}{3}, \\frac{4}{3}, 2, \\frac{8}{3}[\/latex]<\/li>\r\n \t
    4. [latex]0, -5, -10, -15, -20[\/latex]<\/li>\r\n \t
    5. [latex]a_4 = 19[\/latex]<\/li>\r\n \t
    6. [latex]a_6 = 41[\/latex]<\/li>\r\n \t
    7. [latex]a_1 = 2[\/latex]<\/li>\r\n \t
    8. [latex]a_1 = 5[\/latex]<\/li>\r\n \t
    9. [latex]a_1 = 6[\/latex]<\/li>\r\n \t
    10. [latex]a_{21} = -13.5[\/latex]<\/li>\r\n \t
    11. [latex]-19, -20.4, -21.8, -23.2, -24.6[\/latex]<\/li>\r\n \t
    12. [latex]a_1 = 17; a_n = a_{n-1} + 9 \\quad n \\geq 2[\/latex]<\/li>\r\n \t
    13. [latex]a_1 = 12; a_n = a_{n-1} + 5 \\quad n \\geq 2[\/latex]<\/li>\r\n \t
    14. [latex]a_1 = 8.9; a_n = a_{n-1} + 1.4 \\quad n \\geq 2[\/latex]<\/li>\r\n \t
    15. [latex]a_1 = \\frac{1}{5}; a_n = a_{n-1} + \\frac{1}{4} \\quad n \\geq 2[\/latex]<\/li>\r\n \t
    16. [latex]1 = \\frac{1}{6}; a_n = a_{n-1} - \\frac{13}{12} \\quad n \\geq 2[\/latex]<\/li>\r\n \t
    17. [latex]a_1 = 4; a_n = a_{n-1} + 7; a_{14} = 95[\/latex]<\/li>\r\n \t
    18. First five terms: [latex]20, 16, 12, 8, 4[\/latex]<\/li>\r\n \t
    19. [latex]a_n = 1 + 2n[\/latex]<\/li>\r\n \t
    20. [latex]a_n = -105 + 100n[\/latex]<\/li>\r\n \t
    21. [latex]a_n = 1.8n[\/latex]<\/li>\r\n \t
    22. [latex]a_n = 13.1 + 2.7n[\/latex]<\/li>\r\n \t
    23. [latex]a_n = \\frac{1}{3}n - \\frac{1}{3}[\/latex]<\/li>\r\n \t
    24. There are [latex]10[\/latex] terms in the sequence.<\/li>\r\n \t
    25. There are [latex]6[\/latex] terms in the sequence.<\/li>\r\n \t
    26. The graph does not represent an arithmetic sequence.
      <\/center><\/li>\r\n \t
    27. \"Graph<\/li>\r\n<\/ol>\r\n

      Geometric Sequences<\/h2>\r\n
        \r\n \t
      1. The common ratio is [latex]-2[\/latex]<\/li>\r\n \t
      2. The sequence is geometric. The common ratio is [latex]2[\/latex]<\/li>\r\n \t
      3. The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}[\/latex]<\/li>\r\n \t
      4. The sequence is geometric. The common ratio is [latex]5[\/latex]<\/li>\r\n \t
      5. [latex]5, 1, \\frac{1}{5}, \\frac{1}{25}, \\frac{1}{125}[\/latex]<\/li>\r\n \t
      6. [latex]800, 400, 200, 100, 50[\/latex]<\/li>\r\n \t
      7. [latex]a_4 = -\\frac{16}{27}[\/latex]<\/li>\r\n \t
      8. [latex]a_7 = -\\frac{9}{720}[\/latex]<\/li>\r\n \t
      9. [latex]7, 1.4, 0.28, 0.056, 0.0112[\/latex]<\/li>\r\n \t
      10. [latex]a_1 = -32, a_n = \\frac{1}{2}a_{n-1}[\/latex]<\/li>\r\n \t
      11. [latex]a_1 = 10, a_n = -0.3a_{n-1}[\/latex]<\/li>\r\n \t
      12. [latex]a_1 = \\frac{3}{5}, a_n = \\frac{1}{6}a_{n-1}[\/latex]<\/li>\r\n \t
      13. [latex]a_1 = \\frac{1}{512}, a_n = -4a_{n-1}[\/latex]<\/li>\r\n \t
      14. [latex]12, -6, 3, -\\frac{3}{2}, \\frac{3}{4}[\/latex]<\/li>\r\n \t
      15. [latex]a_n = 3^{n-1}[\/latex]<\/li>\r\n \t
      16. [latex]a_n = 0.8 \\cdot (-5)^{n-1}[\/latex]<\/li>\r\n \t
      17. [latex]a_n = -(\\frac{4}{5})^{n-1}[\/latex]<\/li>\r\n \t
      18. [latex]a_n = 3 \\cdot (-\\frac{1}{3})^{n-1}[\/latex]<\/li>\r\n \t
      19. [latex]a_{12} = \\frac{1}{177,147}[\/latex]<\/li>\r\n \t
      20. There are [latex]12[\/latex] terms in the sequence.<\/li>\r\n \t
      21. The graph does not represent a geometric sequence.<\/li>\r\n \t
      22. \"Graph<\/li>\r\n<\/ol>\r\n

        Series and Their Notations<\/h2>\r\n
          \r\n \t
        1. [latex]\\sum_{n=0}^{4} 5n[\/latex]<\/li>\r\n \t
        2. [latex]\\sum_{k=1}^{5} 4[\/latex]<\/li>\r\n \t
        3. [latex]\\sum_{k=1}^{20} 8k + 2[\/latex]<\/li>\r\n \t
        4. [latex]S_5 = \\frac{5(\\frac{3}{2}+\\frac{7}{2})}{2}[\/latex]<\/li>\r\n \t
        5. [latex]S_{13} = \\frac{13(3.2+5.6)}{2}[\/latex]<\/li>\r\n \t
        6. [latex]\\sum_{k=1}^{7} 8 \\cdot 0.5^{k-1}[\/latex]<\/li>\r\n \t
        7. [latex]S_5 = \\frac{9(1-(\\frac{1}{3})^5)}{1-\\frac{1}{3}} = \\frac{121}{9} \\approx 13.44[\/latex]<\/li>\r\n \t
        8. [latex]S_{11} = \\frac{64(1-0.2^{11})}{1-0.2} = \\frac{781,249,984}{9,765,625} \\approx 80[\/latex]<\/li>\r\n \t
        9. The series is defined. [latex]S = \\frac{2}{1-0.8}[\/latex]<\/li>\r\n \t
        10. The series is defined. [latex]S = \\frac{-1}{1-(-\\frac{1}{2})}[\/latex]<\/li>\r\n \t
        11. [latex]49[\/latex]<\/li>\r\n \t
        12. [latex]254[\/latex]<\/li>\r\n \t
        13. [latex]S_7 = \\frac{147}{2}[\/latex]<\/li>\r\n \t
        14. [latex]S_{11} = \\frac{55}{2}[\/latex]<\/li>\r\n \t
        15. [latex]S_7 = 5208.4[\/latex]<\/li>\r\n \t
        16. [latex]S_{10} = -\\frac{1023}{256}[\/latex]<\/li>\r\n \t
        17. [latex]S = -\\frac{4}{3}[\/latex]<\/li>\r\n \t
        18. [latex]S = 9.2[\/latex]<\/li>\r\n \t
        19. [latex]$3,705.42[\/latex]<\/li>\r\n \t
        20. [latex]$695,823.97[\/latex]<\/li>\r\n \t
        21. [latex]420[\/latex] feet<\/li>\r\n \t
        22. [latex]12[\/latex] feet<\/li>\r\n<\/ol>","rendered":"

          Sequences and Their Notations<\/h2>\n
            \n
          1. First four terms: [latex]-8, -\\frac{16}{5}, -4, -\\frac{16}{5}[\/latex]<\/li>\n
          2. First four terms: [latex]2, \\frac{1}{2}, \\frac{8}{27}, \\frac{1}{4}[\/latex]<\/li>\n
          3. First four terms: [latex]1.25, -5, 20, -80[\/latex]<\/li>\n
          4. First four terms: [latex]\\frac{1}{3}, \\frac{4}{5}, \\frac{9}{7}, \\frac{16}{9}[\/latex]<\/li>\n
          5. First four terms: [latex]-\\frac{4}{5}, 4, -20, 100[\/latex]<\/li>\n
          6. [latex]\\frac{1}{3}, \\frac{4}{5}, \\frac{9}{7}, \\frac{16}{9}, \\frac{25}{11}, 31, 44, 59[\/latex]<\/li>\n
          7. [latex]-0.6, -15, -20, -375, -80, -9375, -320[\/latex]<\/li>\n
          8. [latex]a_n = n^2 + 3[\/latex]<\/li>\n
          9. [latex]a_n = \\frac{2^n}{2n}[\/latex] or [latex]2^{n-1}\/n[\/latex]<\/li>\n
          10. [latex]a_n = (-\\frac{1}{2})^{n-1}[\/latex]<\/li>\n
          11. First five terms: [latex]3, -9, 27, -81, 243[\/latex]<\/li>\n
          12. First five terms: [latex]-1, 1, -9, \\frac{27}{11}, \\frac{891}{5}[\/latex]<\/li>\n
          13. [latex]\\frac{1}{24}, 1, \\frac{3}{4}, \\frac{9}{2}, \\frac{81}{4}, \\frac{531,441}{8}, \\frac{16}{16}[\/latex]<\/li>\n
          14. [latex]2, 10, 12, \\frac{14}{5}, \\frac{5}{5}, 2, 10, 12[\/latex]<\/li>\n
          15. [latex]a_1 = -8, a_n = a_{n-1} + n[\/latex]<\/li>\n
          16. [latex]a_1 = 35, a_n = a_{n-1} + 3[\/latex]<\/li>\n
          17. [latex]720[\/latex]<\/li>\n
          18. [latex]665,280[\/latex]<\/li>\n
          19. First four terms: [latex]1, \\frac{1}{2}, \\frac{2}{3}, \\frac{3}{2}[\/latex]<\/li>\n
          20. First four terms: [latex]-1, 2, \\frac{5}{6}, \\frac{24}{11}[\/latex]<\/li>\n
          21. \"Graph<\/li>\n
          22. \"Graph<\/li>\n
          23. \"Graph<\/li>\n
          24. [latex]a_n = 2^{n-2}[\/latex]<\/li>\n
          25. [latex]a_1 = 6, a_n = 2a_{n-1} - 5[\/latex]<\/li>\n<\/ol>\n

            Arithmetic Sequences<\/h2>\n
              \n
            1. The common difference is [latex]\\frac{1}{3}[\/latex]<\/li>\n
            2. The sequence is not arithmetic because [latex]16 - 4 \\neq 64 - 16[\/latex]<\/li>\n
            3. [latex]0, \\frac{2}{3}, \\frac{4}{3}, 2, \\frac{8}{3}[\/latex]<\/li>\n
            4. [latex]0, -5, -10, -15, -20[\/latex]<\/li>\n
            5. [latex]a_4 = 19[\/latex]<\/li>\n
            6. [latex]a_6 = 41[\/latex]<\/li>\n
            7. [latex]a_1 = 2[\/latex]<\/li>\n
            8. [latex]a_1 = 5[\/latex]<\/li>\n
            9. [latex]a_1 = 6[\/latex]<\/li>\n
            10. [latex]a_{21} = -13.5[\/latex]<\/li>\n
            11. [latex]-19, -20.4, -21.8, -23.2, -24.6[\/latex]<\/li>\n
            12. [latex]a_1 = 17; a_n = a_{n-1} + 9 \\quad n \\geq 2[\/latex]<\/li>\n
            13. [latex]a_1 = 12; a_n = a_{n-1} + 5 \\quad n \\geq 2[\/latex]<\/li>\n
            14. [latex]a_1 = 8.9; a_n = a_{n-1} + 1.4 \\quad n \\geq 2[\/latex]<\/li>\n
            15. [latex]a_1 = \\frac{1}{5}; a_n = a_{n-1} + \\frac{1}{4} \\quad n \\geq 2[\/latex]<\/li>\n
            16. [latex]1 = \\frac{1}{6}; a_n = a_{n-1} - \\frac{13}{12} \\quad n \\geq 2[\/latex]<\/li>\n
            17. [latex]a_1 = 4; a_n = a_{n-1} + 7; a_{14} = 95[\/latex]<\/li>\n
            18. First five terms: [latex]20, 16, 12, 8, 4[\/latex]<\/li>\n
            19. [latex]a_n = 1 + 2n[\/latex]<\/li>\n
            20. [latex]a_n = -105 + 100n[\/latex]<\/li>\n
            21. [latex]a_n = 1.8n[\/latex]<\/li>\n
            22. [latex]a_n = 13.1 + 2.7n[\/latex]<\/li>\n
            23. [latex]a_n = \\frac{1}{3}n - \\frac{1}{3}[\/latex]<\/li>\n
            24. There are [latex]10[\/latex] terms in the sequence.<\/li>\n
            25. There are [latex]6[\/latex] terms in the sequence.<\/li>\n
            26. The graph does not represent an arithmetic sequence.\n
              <\/div>\n<\/li>\n
            27. \"Graph<\/li>\n<\/ol>\n

              Geometric Sequences<\/h2>\n
                \n
              1. The common ratio is [latex]-2[\/latex]<\/li>\n
              2. The sequence is geometric. The common ratio is [latex]2[\/latex]<\/li>\n
              3. The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}[\/latex]<\/li>\n
              4. The sequence is geometric. The common ratio is [latex]5[\/latex]<\/li>\n
              5. [latex]5, 1, \\frac{1}{5}, \\frac{1}{25}, \\frac{1}{125}[\/latex]<\/li>\n
              6. [latex]800, 400, 200, 100, 50[\/latex]<\/li>\n
              7. [latex]a_4 = -\\frac{16}{27}[\/latex]<\/li>\n
              8. [latex]a_7 = -\\frac{9}{720}[\/latex]<\/li>\n
              9. [latex]7, 1.4, 0.28, 0.056, 0.0112[\/latex]<\/li>\n
              10. [latex]a_1 = -32, a_n = \\frac{1}{2}a_{n-1}[\/latex]<\/li>\n
              11. [latex]a_1 = 10, a_n = -0.3a_{n-1}[\/latex]<\/li>\n
              12. [latex]a_1 = \\frac{3}{5}, a_n = \\frac{1}{6}a_{n-1}[\/latex]<\/li>\n
              13. [latex]a_1 = \\frac{1}{512}, a_n = -4a_{n-1}[\/latex]<\/li>\n
              14. [latex]12, -6, 3, -\\frac{3}{2}, \\frac{3}{4}[\/latex]<\/li>\n
              15. [latex]a_n = 3^{n-1}[\/latex]<\/li>\n
              16. [latex]a_n = 0.8 \\cdot (-5)^{n-1}[\/latex]<\/li>\n
              17. [latex]a_n = -(\\frac{4}{5})^{n-1}[\/latex]<\/li>\n
              18. [latex]a_n = 3 \\cdot (-\\frac{1}{3})^{n-1}[\/latex]<\/li>\n
              19. [latex]a_{12} = \\frac{1}{177,147}[\/latex]<\/li>\n
              20. There are [latex]12[\/latex] terms in the sequence.<\/li>\n
              21. The graph does not represent a geometric sequence.<\/li>\n
              22. \"Graph<\/li>\n<\/ol>\n

                Series and Their Notations<\/h2>\n
                  \n
                1. [latex]\\sum_{n=0}^{4} 5n[\/latex]<\/li>\n
                2. [latex]\\sum_{k=1}^{5} 4[\/latex]<\/li>\n
                3. [latex]\\sum_{k=1}^{20} 8k + 2[\/latex]<\/li>\n
                4. [latex]S_5 = \\frac{5(\\frac{3}{2}+\\frac{7}{2})}{2}[\/latex]<\/li>\n
                5. [latex]S_{13} = \\frac{13(3.2+5.6)}{2}[\/latex]<\/li>\n
                6. [latex]\\sum_{k=1}^{7} 8 \\cdot 0.5^{k-1}[\/latex]<\/li>\n
                7. [latex]S_5 = \\frac{9(1-(\\frac{1}{3})^5)}{1-\\frac{1}{3}} = \\frac{121}{9} \\approx 13.44[\/latex]<\/li>\n
                8. [latex]S_{11} = \\frac{64(1-0.2^{11})}{1-0.2} = \\frac{781,249,984}{9,765,625} \\approx 80[\/latex]<\/li>\n
                9. The series is defined. [latex]S = \\frac{2}{1-0.8}[\/latex]<\/li>\n
                10. The series is defined. [latex]S = \\frac{-1}{1-(-\\frac{1}{2})}[\/latex]<\/li>\n
                11. [latex]49[\/latex]<\/li>\n
                12. [latex]254[\/latex]<\/li>\n
                13. [latex]S_7 = \\frac{147}{2}[\/latex]<\/li>\n
                14. [latex]S_{11} = \\frac{55}{2}[\/latex]<\/li>\n
                15. [latex]S_7 = 5208.4[\/latex]<\/li>\n
                16. [latex]S_{10} = -\\frac{1023}{256}[\/latex]<\/li>\n
                17. [latex]S = -\\frac{4}{3}[\/latex]<\/li>\n
                18. [latex]S = 9.2[\/latex]<\/li>\n
                19. [latex]$3,705.42[\/latex]<\/li>\n
                20. [latex]$695,823.97[\/latex]<\/li>\n
                21. [latex]420[\/latex] feet<\/li>\n
                22. [latex]12[\/latex] feet<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":106,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/137"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions"}],"predecessor-version":[{"id":145,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions\/145"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/106"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}