{"id":2765,"date":"2024-08-16T00:09:10","date_gmt":"2024-08-16T00:09:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2765"},"modified":"2024-12-02T19:14:45","modified_gmt":"2024-12-02T19:14:45","slug":"series-and-their-notations-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/series-and-their-notations-apply-it-2\/","title":{"raw":"Series and Their Notations: Apply It 2","rendered":"Series and Their Notations: Apply It 2"},"content":{"raw":"

Annuities<\/h2>\r\nAn annuity<\/strong> is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned.\r\n\r\n
A parent invests [latex]$50[\/latex] each month into a college fund. The account paid [latex]6\\%[\/latex] annual interest<\/strong>, compounded monthly. How much money will be accumulated in [latex]6[\/latex] years?\r\n\r\n
\r\n\r\nTo find the interest rate per payment period, we need to divide the [latex]6\\%[\/latex] annual percentage interest (APR) rate by [latex]12[\/latex]. So, the monthly interest rate is [latex]0.5\\%[\/latex]. We can multiply the amount in the account each month by [latex]100.5 \\%[\/latex] to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5 \\%=1.005[\/latex].\r\n
    \r\n \t
  • After the first deposit, the value of the annuity will be [latex]$50[\/latex].<\/li>\r\n<\/ul>\r\nLet us see if we can determine the amount in the college fund and the interest earned. We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In [latex]6[\/latex] years, there are [latex]72[\/latex] months, so [latex]n=72[\/latex]. We can substitute [latex]{a}_{1}=50, r=1.005,[\/latex] and [latex]n=72[\/latex] into the formula, and simplify to find the value of the annuity after [latex]6[\/latex] years.\r\n

    [latex]{S}_{72}=\\dfrac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/p>\r\nAfter the last deposit, the couple will have a total of [latex]$4,320.44[\/latex] in the account. <\/strong>\r\n\r\nNotice, the couple made [latex]72[\/latex] payments of [latex]$50[\/latex] each for a total of [latex]72\\left(50\\right) = $3,600[\/latex].\r\n\r\nThis means that because of the annuity, the couple earned [latex]$720.44[\/latex] interest in their college fund.\r\n\r\n<\/section>

    How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/strong>\r\n
      \r\n \t
    1. Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\r\n \t
    2. Determine [latex]n[\/latex], the number of deposits.<\/li>\r\n \t
    3. Determine [latex]r[\/latex].\r\n
        \r\n \t
      1. Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\r\n \t
      2. Add 1 to this amount to find [latex]r[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      3. Substitute values for [latex]{a}_{1},r,[\/latex] and [latex]n[\/latex]\r\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\r\n \t
      4. Simplify to find [latex]{S}_{n}[\/latex], the value of the annuity after [latex]n[\/latex] deposits.<\/li>\r\n<\/ol>\r\n<\/section>
        A deposit of [latex]$100[\/latex] is placed into a college fund at the beginning of every month for [latex]10[\/latex] years. The fund earns [latex]9 \\%[\/latex] annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?[reveal-answer q=\"808488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808488\"]The value of the initial deposit is [latex]$100[\/latex], so [latex]{a}_{1}=100[\/latex]. A total of [latex]120[\/latex] monthly deposits are made in the [latex]10[\/latex] years, so [latex]n=120[\/latex]. To find [latex]r[\/latex], divide the annual interest rate by [latex]12[\/latex] to find the monthly interest rate and add [latex]1[\/latex] to represent the new monthly deposit.\r\n

        [latex]r=1+\\dfrac{0.09}{12}=1.0075[\/latex]<\/p>\r\nSubstitute [latex]{a}_{1}=100,r=1.0075,[\/latex] and [latex]n=120[\/latex] into the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, and simplify to find the value of the annuity.\r\n

        [latex]{S}_{120}=\\dfrac{100\\left(1-{1.0075}^{120}\\right)}{1 - 1.0075}\\approx 19\\text{,}351.43[\/latex]<\/p>\r\nSo the account has [latex]$19,351.43[\/latex] after the last deposit is made.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        At the beginning of each month, [latex]$200[\/latex] is deposited into a retirement fund. The fund earns [latex]6 \\%[\/latex] annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for [latex]10[\/latex] years?[reveal-answer q=\"786342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786342\"][latex]$92,408.18[\/latex][\/hidden-answer]<\/section>
        [ohm2_question hide_question_numbers=1]24958[\/ohm2_question]<\/section>","rendered":"

        Annuities<\/h2>\n

        An annuity<\/strong> is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned.<\/p>\n

        A parent invests [latex]$50[\/latex] each month into a college fund. The account paid [latex]6\\%[\/latex] annual interest<\/strong>, compounded monthly. How much money will be accumulated in [latex]6[\/latex] years?<\/p>\n
        \n

        To find the interest rate per payment period, we need to divide the [latex]6\\%[\/latex] annual percentage interest (APR) rate by [latex]12[\/latex]. So, the monthly interest rate is [latex]0.5\\%[\/latex]. We can multiply the amount in the account each month by [latex]100.5 \\%[\/latex] to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5 \\%=1.005[\/latex].<\/p>\n

          \n
        • After the first deposit, the value of the annuity will be [latex]$50[\/latex].<\/li>\n<\/ul>\n

          Let us see if we can determine the amount in the college fund and the interest earned. We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In [latex]6[\/latex] years, there are [latex]72[\/latex] months, so [latex]n=72[\/latex]. We can substitute [latex]{a}_{1}=50, r=1.005,[\/latex] and [latex]n=72[\/latex] into the formula, and simplify to find the value of the annuity after [latex]6[\/latex] years.<\/p>\n

          [latex]{S}_{72}=\\dfrac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/p>\n

          After the last deposit, the couple will have a total of [latex]$4,320.44[\/latex] in the account. <\/strong><\/p>\n

          Notice, the couple made [latex]72[\/latex] payments of [latex]$50[\/latex] each for a total of [latex]72\\left(50\\right) = $3,600[\/latex].<\/p>\n

          This means that because of the annuity, the couple earned [latex]$720.44[\/latex] interest in their college fund.<\/p>\n<\/section>\n

          How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/strong><\/p>\n
            \n
          1. Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\n
          2. Determine [latex]n[\/latex], the number of deposits.<\/li>\n
          3. Determine [latex]r[\/latex].\n
              \n
            1. Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\n
            2. Add 1 to this amount to find [latex]r[\/latex].<\/li>\n<\/ol>\n<\/li>\n
            3. Substitute values for [latex]{a}_{1},r,[\/latex] and [latex]n[\/latex]
              \ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\n
            4. Simplify to find [latex]{S}_{n}[\/latex], the value of the annuity after [latex]n[\/latex] deposits.<\/li>\n<\/ol>\n<\/section>\n
              A deposit of [latex]$100[\/latex] is placed into a college fund at the beginning of every month for [latex]10[\/latex] years. The fund earns [latex]9 \\%[\/latex] annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?<\/p>\n