{"id":2058,"date":"2015-11-12T18:30:42","date_gmt":"2015-11-12T18:30:42","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2058"},"modified":"2015-11-12T18:30:42","modified_gmt":"2015-11-12T18:30:42","slug":"solving-application-problems-with-geometric-sequences","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solving-application-problems-with-geometric-sequences\/","title":{"raw":"Solving Application Problems with Geometric Sequences","rendered":"Solving Application Problems with Geometric Sequences"},"content":{"raw":"

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:\n<\/p>

[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\n
\n

Example 7: Solving Application Problems with Geometric Sequences<\/h3>\nIn 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.\n
  1. Write a formula for the student population.<\/li>\n\t
  2. Estimate the student population in 2020.<\/li>\n<\/ol><\/div>\n
    \n

    Solution<\/h3>\n
    1. The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.\n\nLet [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get\n
      [latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div><\/li>\n\t
    2. We can find the number of years since 2013 by subtracting.\n
      [latex]2020 - 2013=7[\/latex]<\/div>\nWe are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.\n
      [latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\nThe student population will be about 374 in 2020.<\/li>\n<\/ol><\/div>\n
      \n

      Try It 7<\/h3>\nA business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.\n
      \n
      a. Write a formula for the number of hits.<\/div>\n
      b. Estimate the number of hits in 5 weeks.<\/div>\n
      Solution<\/a><\/div>\n\n<\/div>\n<\/div>","rendered":"

      In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:\n<\/p>\n

      [latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\n
      \n

      Example 7: Solving Application Problems with Geometric Sequences<\/h3>\n

      In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.<\/p>\n

        \n
      1. Write a formula for the student population.<\/li>\n
      2. Estimate the student population in 2020.<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n
          \n
        1. The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.\n

          Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get<\/p>\n

          [latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div>\n<\/li>\n
        2. We can find the number of years since 2013 by subtracting.\n
          [latex]2020 - 2013=7[\/latex]<\/div>\n

          We are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.<\/p>\n

          [latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\n

          The student population will be about 374 in 2020.<\/li>\n<\/ol>\n<\/div>\n

          \n

          Try It 7<\/h3>\n

          A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.<\/p>\n

          \n
          a. Write a formula for the number of hits.<\/div>\n
          b. Estimate the number of hits in 5 weeks.<\/div>\n
          Solution<\/a><\/div>\n
          \n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
          \n\t\t\t

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