{"id":5242,"date":"2018-05-12T18:17:41","date_gmt":"2018-05-12T18:17:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/coreq-collegealgebra\/?post_type=chapter&p=5242"},"modified":"2018-05-17T04:57:49","modified_gmt":"2018-05-17T04:57:49","slug":"why-it-matters-sequences-and-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/why-it-matters-sequences-and-series\/","title":{"raw":"Why It Matters: Sequences and Series*","rendered":"Why It Matters: Sequences and Series*"},"content":{"raw":"

Why Study Sequences and Series?<\/h2>\r\nA sequence<\/strong> is simply a list of numbers, and a series<\/strong> is the sum of a list of numbers. \u00a0So any time you have data arranged in a list, you may require methods from sequences and series to analyze the data.\r\n\r\nFor example, suppose you take out a small student loan for $10,000. \u00a0When the loan is in repayment, it will accrue 6% annual interest, compounded each month. \u00a0Suppose that the monthly payments were fixed at $300 by the lender. \u00a0How long will it take to pay off the loan? \u00a0How much interest will you end up paying?\r\n\r\nTo answer these questions quickly, we would have to know about certain loan formulas. \u00a0However we can still answer the questions by finding out exactly what is still left to pay after each month. \u00a0That is, we will examine the sequence of balances<\/strong>.\r\n\r\nLet\u2019s work out the first month carefully. \u00a0First of all, the bank will tack on interest. \u00a0Each month, the bank charges 0.5% interest [latex](6\\% \\div 12)[\/latex], or as a decimal: 0.005. \u00a0Let [latex]I_n[\/latex] stand for the interest payment in month [latex]n[\/latex].\r\n

[latex]I_1 = \\$10,\\!000 \\times 0.005 = \\$50[\/latex]<\/p>\r\nTherefore out of the first $300, only $250 will go towards paying off the loan. \u00a0Let [latex]A_n[\/latex] be the balance after month [latex]n[\/latex]. \u00a0By convention, we write [latex]A_0 = 10,\\!000[\/latex] for the initial amount of the loan (let\u2019s not worry about the dollar signs going forward). \u00a0So after the first month, the new balance would be:\r\n

[latex]A_1 = 10,\\!000 - 250 = 9,\\!750[\/latex]<\/p>\r\nTo compute the balance after the second month, we repeat the same steps. \u00a0The new interest payment is:\r\n

[latex]I_2 = 9750 \\times 0.005 = 48.75[\/latex]<\/p>\r\nThat would leave [latex]300-48.75=251.25[\/latex] for paying off the loan. \u00a0Altogether then, the new balance will be:\r\n

[latex]A_2 = 9750 - 251.25 = 9498.75[\/latex]<\/p>\r\nBut now in order to automate the process, let\u2019s write everything in terms of [latex]I_n[\/latex] and [latex]A_n[\/latex]. \u00a0First, the interest payment is always equal to the previous balance times 0.005.\r\n

[latex]I_n = A_{n-1} \\times 0.005[\/latex]<\/p>\r\nThen the new payment is [latex]300 - I_n[\/latex]. \u00a0This leads to the formula for the new balance (which we can simplify algebraically):\r\n

[latex]A_n = A_{n-1} - (300 - I_n) = A_{n-1} - 300 + I_n [\/latex]<\/p>\r\n

[latex]= A_{n-1} - 300 + A_{n-1} \\times (0.005)[\/latex]<\/p>\r\n

[latex]= (1.005)A_{n-1} - 300[\/latex]<\/p>\r\nAlthough it seems complicated at first, the formula for [latex]A_n[\/latex], which is called a recursive<\/strong> definition for the sequence, helps us to compute all of the balances very quickly. \u00a0The reason is that each balance calculation is found by a simple formula from the previous. \u00a0In fact, spreadsheets are very good at this kind of work. \u00a0Here is a table of the first six months of balances and interest payments.\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Month<\/td>\r\nBalance<\/td>\r\nInterest Paid<\/td>\r\n<\/tr>\r\n
0<\/td>\r\n$10,000<\/td>\r\n-<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n$9750<\/td>\r\n$50<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n$9498.75<\/td>\r\n$48.75<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n$9246.24<\/td>\r\n$47.49<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n$8992.47<\/td>\r\n$46.23<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n$8737.43<\/td>\r\n$44.96<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n$8481.12<\/td>\r\n$43.69<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nTry filling out more of the table. \u00a0You may find that after about 3 years, the student loan will be paid off.\r\n\r\nHow much interest did you pay over the life of the loan? \u00a0In this module you will also learn about sums of sequences<\/strong>, or series<\/strong>. \u00a0Just add the terms of the series [latex]I_1 + I_2 + I_3 + \\cdots[\/latex] in order to figure out how much total interest was paid. \u00a0That 6% interest really adds up, doesn\u2019t it?\r\n
\r\n

Learning Objectives<\/h3>\r\nSequences and Their Notations\r\n
    \r\n \t
  • Write the terms of a sequence defined by an explicit formula<\/span><\/li>\r\n \t
  • Write the terms of a sequence defined by a recursive formula<\/span><\/li>\r\n \t
  • Use factorial notation<\/span><\/li>\r\n<\/ul>\r\n

    \u00a0Arithmetic Sequences<\/span><\/p>\r\n\r\n

      \r\n \t
    • Find the common difference for an arithmetic sequence<\/span><\/li>\r\n \t
    • Write terms of an arithmetic sequence<\/span><\/li>\r\n \t
    • Use a recursive formula for an arithmetic sequence<\/span><\/li>\r\n \t
    • Use an explicit formula for an arithmetic sequence<\/span><\/li>\r\n<\/ul>\r\n

      Geometric Sequences<\/span><\/p>\r\n\r\n

        \r\n \t
      • Find the common ratio for a geometric sequence<\/span><\/li>\r\n \t
      • List the terms of a geometric sequence<\/span><\/li>\r\n \t
      • Use a recursive formula for a geometric sequence<\/span><\/li>\r\n \t
      • Use an explicit formula for a geometric sequence<\/span><\/li>\r\n<\/ul>\r\n

        Series and Their Notations<\/span><\/p>\r\n\r\n

          \r\n \t
        • Use summation notation<\/span><\/li>\r\n \t
        • Use the formula for the sum of the \ufb01rst <\/span>n\u00a0<\/i><\/span>terms of an arithmetic series<\/span><\/li>\r\n \t
        • Use the formula for the sum of the \ufb01rst <\/span>n\u00a0<\/i><\/span>terms of a geometric series<\/span><\/li>\r\n \t
        • Use the formula for the sum of an in\ufb01nite geometric series<\/span><\/li>\r\n \t
        • Solve annuity problems<\/span><\/li>\r\n<\/ul>\r\n<\/div>","rendered":"

          Why Study Sequences and Series?<\/h2>\n

          A sequence<\/strong> is simply a list of numbers, and a series<\/strong> is the sum of a list of numbers. \u00a0So any time you have data arranged in a list, you may require methods from sequences and series to analyze the data.<\/p>\n

          For example, suppose you take out a small student loan for $10,000. \u00a0When the loan is in repayment, it will accrue 6% annual interest, compounded each month. \u00a0Suppose that the monthly payments were fixed at $300 by the lender. \u00a0How long will it take to pay off the loan? \u00a0How much interest will you end up paying?<\/p>\n

          To answer these questions quickly, we would have to know about certain loan formulas. \u00a0However we can still answer the questions by finding out exactly what is still left to pay after each month. \u00a0That is, we will examine the sequence of balances<\/strong>.<\/p>\n

          Let\u2019s work out the first month carefully. \u00a0First of all, the bank will tack on interest. \u00a0Each month, the bank charges 0.5% interest [latex](6\\% \\div 12)[\/latex], or as a decimal: 0.005. \u00a0Let [latex]I_n[\/latex] stand for the interest payment in month [latex]n[\/latex].<\/p>\n

          [latex]I_1 = \\$10,\\!000 \\times 0.005 = \\$50[\/latex]<\/p>\n

          Therefore out of the first $300, only $250 will go towards paying off the loan. \u00a0Let [latex]A_n[\/latex] be the balance after month [latex]n[\/latex]. \u00a0By convention, we write [latex]A_0 = 10,\\!000[\/latex] for the initial amount of the loan (let\u2019s not worry about the dollar signs going forward). \u00a0So after the first month, the new balance would be:<\/p>\n

          [latex]A_1 = 10,\\!000 - 250 = 9,\\!750[\/latex]<\/p>\n

          To compute the balance after the second month, we repeat the same steps. \u00a0The new interest payment is:<\/p>\n

          [latex]I_2 = 9750 \\times 0.005 = 48.75[\/latex]<\/p>\n

          That would leave [latex]300-48.75=251.25[\/latex] for paying off the loan. \u00a0Altogether then, the new balance will be:<\/p>\n

          [latex]A_2 = 9750 - 251.25 = 9498.75[\/latex]<\/p>\n

          But now in order to automate the process, let\u2019s write everything in terms of [latex]I_n[\/latex] and [latex]A_n[\/latex]. \u00a0First, the interest payment is always equal to the previous balance times 0.005.<\/p>\n

          [latex]I_n = A_{n-1} \\times 0.005[\/latex]<\/p>\n

          Then the new payment is [latex]300 - I_n[\/latex]. \u00a0This leads to the formula for the new balance (which we can simplify algebraically):<\/p>\n

          [latex]A_n = A_{n-1} - (300 - I_n) = A_{n-1} - 300 + I_n[\/latex]<\/p>\n

          [latex]= A_{n-1} - 300 + A_{n-1} \\times (0.005)[\/latex]<\/p>\n

          [latex]= (1.005)A_{n-1} - 300[\/latex]<\/p>\n

          Although it seems complicated at first, the formula for [latex]A_n[\/latex], which is called a recursive<\/strong> definition for the sequence, helps us to compute all of the balances very quickly. \u00a0The reason is that each balance calculation is found by a simple formula from the previous. \u00a0In fact, spreadsheets are very good at this kind of work. \u00a0Here is a table of the first six months of balances and interest payments.<\/p>\n

          \n\n\n\n\n\n\n\n\n\n\n
          Month<\/td>\nBalance<\/td>\nInterest Paid<\/td>\n<\/tr>\n
          0<\/td>\n$10,000<\/td>\n–<\/td>\n<\/tr>\n
          1<\/td>\n$9750<\/td>\n$50<\/td>\n<\/tr>\n
          2<\/td>\n$9498.75<\/td>\n$48.75<\/td>\n<\/tr>\n
          3<\/td>\n$9246.24<\/td>\n$47.49<\/td>\n<\/tr>\n
          4<\/td>\n$8992.47<\/td>\n$46.23<\/td>\n<\/tr>\n
          5<\/td>\n$8737.43<\/td>\n$44.96<\/td>\n<\/tr>\n
          6<\/td>\n$8481.12<\/td>\n$43.69<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

          Try filling out more of the table. \u00a0You may find that after about 3 years, the student loan will be paid off.<\/p>\n

          How much interest did you pay over the life of the loan? \u00a0In this module you will also learn about sums of sequences<\/strong>, or series<\/strong>. \u00a0Just add the terms of the series [latex]I_1 + I_2 + I_3 + \\cdots[\/latex] in order to figure out how much total interest was paid. \u00a0That 6% interest really adds up, doesn\u2019t it?<\/p>\n

          \n

          Learning Objectives<\/h3>\n

          Sequences and Their Notations<\/p>\n

            \n
          • Write the terms of a sequence defined by an explicit formula<\/span><\/li>\n
          • Write the terms of a sequence defined by a recursive formula<\/span><\/li>\n
          • Use factorial notation<\/span><\/li>\n<\/ul>\n

            \u00a0Arithmetic Sequences<\/span><\/p>\n

              \n
            • Find the common difference for an arithmetic sequence<\/span><\/li>\n
            • Write terms of an arithmetic sequence<\/span><\/li>\n
            • Use a recursive formula for an arithmetic sequence<\/span><\/li>\n
            • Use an explicit formula for an arithmetic sequence<\/span><\/li>\n<\/ul>\n

              Geometric Sequences<\/span><\/p>\n

                \n
              • Find the common ratio for a geometric sequence<\/span><\/li>\n
              • List the terms of a geometric sequence<\/span><\/li>\n
              • Use a recursive formula for a geometric sequence<\/span><\/li>\n
              • Use an explicit formula for a geometric sequence<\/span><\/li>\n<\/ul>\n

                Series and Their Notations<\/span><\/p>\n

                  \n
                • Use summation notation<\/span><\/li>\n
                • Use the formula for the sum of the \ufb01rst <\/span>n\u00a0<\/i><\/span>terms of an arithmetic series<\/span><\/li>\n
                • Use the formula for the sum of the \ufb01rst <\/span>n\u00a0<\/i><\/span>terms of a geometric series<\/span><\/li>\n
                • Use the formula for the sum of an in\ufb01nite geometric series<\/span><\/li>\n
                • Solve annuity problems<\/span><\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":160,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5242","chapter","type-chapter","status-publish","hentry"],"part":473,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5242","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/160"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5242\/revisions"}],"predecessor-version":[{"id":5243,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5242\/revisions\/5243"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/473"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5242\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=5242"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=5242"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=5242"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=5242"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}