{"id":1681,"date":"2025-07-25T19:11:44","date_gmt":"2025-07-25T19:11:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1681"},"modified":"2026-03-25T15:38:38","modified_gmt":"2026-03-25T15:38:38","slug":"sequences-and-their-notations-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sequences-and-their-notations-learn-it-3\/","title":{"raw":"Sequences and Their Notations: Learn It 3","rendered":"Sequences and Their Notations: Learn It 3"},"content":{"raw":"

Recursive Formula<\/h2>\r\nNow that we\u2019ve covered explicit formulas, let\u2019s talk about another way to define sequences: the recursive formula<\/strong>. Unlike the explicit formula, which gives you a direct way to find any term in the sequence, a recursive formula builds each term based on the one before it. In other words, you need to know the previous term(s) to find the next one.\r\n\r\n
\r\n

recursive formula<\/h3>\r\nA recursive formula<\/strong> is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.\r\n\r\n<\/section>Recursive formulas are especially useful when the sequence is defined by a step-by-step process or when each term depends on its predecessor. While it might seem a bit more complex, recursive formulas can be powerful tools for understanding and working with sequences.\r\n\r\n
Do you notice how the sequence seems to mirror patterns found in nature?\"What\r\n[latex]\\\\[\/latex]\r\nThe Fibonacci sequence<\/strong> isn\u2019t just a mathematical curiosity\u2014it\u2019s a fundamental part of the world around us. From the spirals of shells to the branching of trees, this sequence helps explain the structure and growth patterns of various natural forms. What\u2019s fascinating is that each term relies on the ones before it, creating a beautiful and interconnected pattern that we can observe in so many living things. This dependency makes the Fibonacci sequence unique, as it\u2019s best described by a recursive formula rather than an explicit one.\r\n[latex]\\\\[\/latex]\r\nFibonacci sequence is [latex]1, 1, 2, 3, 5, 8, 13, 21, 34,\u2026.[\/latex].\r\n[latex]\\\\[\/latex]\r\nEach number in the sequence is the sum of the two preceding ones, starting from [latex]1[\/latex] and [latex]1[\/latex]. This sequence is unique because of its recursive nature, meaning each term is defined by the terms that came before it.The formula for the Fibonacci sequence is:\r\n
    \r\n \t
  • [latex]F_1 = 1[\/latex]<\/li>\r\n \t
  • [latex]F_2 = 1[\/latex]<\/li>\r\n \t
  • [latex]F_n = F_{n-1}+F_{n-2}[\/latex], for [latex]n \\ge 3[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>
    How To: Given a recursive formula with only the first term provided, write the first [latex]n[\/latex] terms of a sequence.<\/strong>\r\n
      \r\n \t
    1. Identify the initial term, [latex]{a}_{1}[\/latex], which is given as part of the formula. This is the first term.<\/li>\r\n \t
    2. To find the second term, [latex]{a}_{2}[\/latex], substitute the initial term into the formula for [latex]{a}_{n - 1}[\/latex]. Solve.<\/li>\r\n \t
    3. To find the third term, [latex]{a}_{3}[\/latex], substitute the second term into the formula. Solve.<\/li>\r\n \t
    4. Repeat until you have solved for the [latex]n\\text{th}[\/latex] term.<\/li>\r\n<\/ol>\r\n<\/section>
      Write the first five terms of the sequence defined by the recursive formula.\r\n

      [latex]\\begin{align} {a}_{1}&=9 \\\\ {a}_{n}&=3{a}_{n - 1}-20\\text{, for }n\\ge 2 \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"748916\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"748916\"]\r\n\r\nThe first term is given in the formula. For each subsequent term, we replace [latex]{a}_{n - 1}[\/latex] with the value of the preceding term.\r\n

      [latex]\\begin{align}&n=1 && {a}_{1}=9 \\\\ &n=2 && {a}_{2}=3{a}_{1}-20=3\\left(9\\right)-20=27 - 20=7 \\\\ &n=3 && {a}_{3}=3{a}_{2}-20=3\\left(7\\right)-20=21 - 20=1 \\\\ &n=4 && {a}_{4}=3{a}_{3}-20=3\\left(1\\right)-20=3 - 20=-17 \\\\ &n=5 && {a}_{5}=3{a}_{4}-20=3\\left(-17\\right)-20=-51 - 20=-71 \\end{align}[\/latex]<\/p>\r\nThe first five terms are [latex]\\left\\{9,7,1,-17,-71\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      [ohm_question hide_question_numbers=1]321874[\/ohm_question]<\/section>","rendered":"

      Recursive Formula<\/h2>\n

      Now that we\u2019ve covered explicit formulas, let\u2019s talk about another way to define sequences: the recursive formula<\/strong>. Unlike the explicit formula, which gives you a direct way to find any term in the sequence, a recursive formula builds each term based on the one before it. In other words, you need to know the previous term(s) to find the next one.<\/p>\n

      \n

      recursive formula<\/h3>\n

      A recursive formula<\/strong> is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.<\/p>\n<\/section>\n

      Recursive formulas are especially useful when the sequence is defined by a step-by-step process or when each term depends on its predecessor. While it might seem a bit more complex, recursive formulas can be powerful tools for understanding and working with sequences.<\/p>\n

      Do you notice how the sequence seems to mirror patterns found in nature?\"What
      \n[latex]\\\\[\/latex]
      \nThe Fibonacci sequence<\/strong> isn\u2019t just a mathematical curiosity\u2014it\u2019s a fundamental part of the world around us. From the spirals of shells to the branching of trees, this sequence helps explain the structure and growth patterns of various natural forms. What\u2019s fascinating is that each term relies on the ones before it, creating a beautiful and interconnected pattern that we can observe in so many living things. This dependency makes the Fibonacci sequence unique, as it\u2019s best described by a recursive formula rather than an explicit one.
      \n[latex]\\\\[\/latex]
      \nFibonacci sequence is [latex]1, 1, 2, 3, 5, 8, 13, 21, 34,\u2026.[\/latex].
      \n[latex]\\\\[\/latex]
      \nEach number in the sequence is the sum of the two preceding ones, starting from [latex]1[\/latex] and [latex]1[\/latex]. This sequence is unique because of its recursive nature, meaning each term is defined by the terms that came before it.The formula for the Fibonacci sequence is:<\/p>\n
        \n
      • [latex]F_1 = 1[\/latex]<\/li>\n
      • [latex]F_2 = 1[\/latex]<\/li>\n
      • [latex]F_n = F_{n-1}+F_{n-2}[\/latex], for [latex]n \\ge 3[\/latex]<\/li>\n<\/ul>\n<\/section>\n
        How To: Given a recursive formula with only the first term provided, write the first [latex]n[\/latex] terms of a sequence.<\/strong><\/p>\n
          \n
        1. Identify the initial term, [latex]{a}_{1}[\/latex], which is given as part of the formula. This is the first term.<\/li>\n
        2. To find the second term, [latex]{a}_{2}[\/latex], substitute the initial term into the formula for [latex]{a}_{n - 1}[\/latex]. Solve.<\/li>\n
        3. To find the third term, [latex]{a}_{3}[\/latex], substitute the second term into the formula. Solve.<\/li>\n
        4. Repeat until you have solved for the [latex]n\\text{th}[\/latex] term.<\/li>\n<\/ol>\n<\/section>\n
          Write the first five terms of the sequence defined by the recursive formula.<\/p>\n

          [latex]\\begin{align} {a}_{1}&=9 \\\\ {a}_{n}&=3{a}_{n - 1}-20\\text{, for }n\\ge 2 \\end{align}[\/latex]<\/p>\n