{"id":1485,"date":"2023-04-10T15:59:04","date_gmt":"2023-04-10T15:59:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1485"},"modified":"2024-10-18T20:53:51","modified_gmt":"2024-10-18T20:53:51","slug":"fractals-generated-by-complex-numbers-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/fractals-generated-by-complex-numbers-learn-it-4\/","title":{"raw":"Fractals Generated by Complex Numbers: Learn It 4","rendered":"Fractals Generated by Complex Numbers: Learn It 4"},"content":{"raw":"

Complex Recursive Sequences<\/h2>\r\n

We will now explore recursively defined sequences of complex numbers. Recursively defined sequences are sequences in which subsequent terms are constructed based on preceding terms using a specific set of rules or formulas, allowing each term to be defined as a function of its predecessors.<\/p>\r\n

\r\n
\r\n

Recursive Sequence<\/h3>\r\n

A recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].<\/p>\r\n

 <\/p>\r\n

The sequence of values produced is the recursive sequence<\/strong>.<\/p>\r\n<\/div>\r\n<\/section>\r\n

In mathematics, we often use subscripts to denote specific elements or terms in a sequence or set. Imagine you have a list of data entries represented by the symbol [latex]x[\/latex]. These entries can be labeled as [latex]x_1,x_2,x_3, ...[\/latex], where the subscript indicates the position of the entry in the list.\r\n\r\n\r\n

For instance, [latex]x_1[\/latex] refers to the first entry, and [latex]x_2[\/latex] refers to the second entry, and so on. We sometimes start our list with a \"zeroth\" term, denoted as [latex]x_0[\/latex]. When we want to talk about a general term in the list, we use [latex]x_n[\/latex], where [latex]n[\/latex] represents any position in the list. This notation also allows us to refer to the terms before and after [latex]x_n[\/latex] as [latex]x_{n-1}[\/latex] and [latex]x_{n+1}[\/latex], respectively.<\/p>\r\n<\/section>\r\n

\r\n

How To: Apply a Recursive Sequence<\/strong><\/p>\r\n

    \r\n\t
  1. Identify the Initial Term:<\/strong> Start by determining the first term of the sequence, often denoted as [latex]a_0[\/latex] or [latex]a_1[\/latex], provided in the sequence definition. This term serves as the starting point for building the rest of the sequence.<\/li>\r\n\t
  2. Understand the Recursive Formula:<\/strong> Look at the recursive relationship that defines how to find each term from the previous term(s). The formula will generally be given in the form of [latex]a_{n+1}=f(a_n)[\/latex], where [latex]f[\/latex]\u00a0represents a function.<\/li>\r\n\t
  3. Apply the Formula:<\/strong> Use the formula to calculate the next term in the sequence by substituting the previous term into the formula.<\/li>\r\n\t
  4. Repeat the Process:<\/strong> Continue applying the recursive formula to each new term to find subsequent terms as needed.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
    Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2, {{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.
    \r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"703380\"]
    \r\nWe are given the starting value, [latex]{{z}_{0}}=4[\/latex]. The recursive formula holds for any value of [latex]n[\/latex], so if [latex]n = 0[\/latex], then [latex]{{z}_{n+1}}={{z}_{n}}+2[\/latex] would tell us [latex]{{z}_{0+1}}={{z}_{0}}+2[\/latex], or more simply, [latex]{{z}_{1}}={{z}_{0}}+2[\/latex]. Notice this defines [latex]{{z}_{1}}[\/latex] in terms of the known [latex]{{z}_{0}}[\/latex], so we can compute the value:\r\n\r\n\r\n

    [latex]{{z}_{1}}={{z}_{0}}+2=4+2=6[\/latex].<\/p>\r\n

    Now letting [latex]n = 1[\/latex], the formula tells us [latex]{{z}_{1+1}}={{z}_{1}}+2[\/latex], or [latex]{{z}_{2}}={{z}_{1}}+2[\/latex]. Again, the formula gives the next value in the sequence in terms of the previous value.<\/p>\r\n

    [latex]{{z}_{2}}={{z}_{1}}+2=6+2=8[\/latex]<\/p>\r\n

    Continuing,<\/p>\r\n

    [latex]{{z}_{3}}={{z}_{2}}+2=8+2=10[\/latex]<\/center>
    [latex]{{z}_{4}}={{z}_{3}}+2=10+2=12[\/latex]<\/center>\r\n


    \r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n

    [ohm2_question hide_question_numbers=1]6897[\/ohm2_question]<\/section>\r\n

    The previous example generated a basic linear sequence of real numbers. The same process can be used with complex numbers.<\/p>\r\n

    Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}\\cdot{i}+(1-i), {{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.
    \r\n[reveal-answer q=\"703381\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"703381\"]\r\n\r\n\r\n

    We are given [latex]{{z}_{0}}=4[\/latex]. Using the recursive formula:<\/p>\r\n

    [latex]\\begin{array}{l}\r\nz_1 = z_0 \\cdot i + (1 - i) = 4 \\cdot i + (1 - i) = 1 + 3i \\\\\r\nz_2 = z_1 \\cdot i + (1 - i) = (1 + 3i) \\cdot i + (1 - i) = i + 3i^2 + (1 - i) = i - 3 + (1 - i) = -2 \\\\\r\nz_3 = z_2 \\cdot i + (1 - i) = (-2) \\cdot i + (1 - i) = -2i + (1 - i) = 1 - 3i \\\\\r\nz_4 = z_3 \\cdot i + (1 - i) = (1 - 3i) \\cdot i + (1 - i) = i - 3i^2 + (1 - i) = i + 3 + (1 - i) = 4 \\\\\r\nz_5 = z_4 \\cdot i + (1 - i) = 4 \\cdot i + (1 - i) = 1 + 3i\r\n\\end{array}\r\n[\/latex]<\/p>\r\n

    Notice this sequence is exhibiting an interesting pattern\u2014it began to repeat itself.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n

    [ohm2_question hide_question_numbers=1]6898[\/ohm2_question]<\/section>","rendered":"

    Complex Recursive Sequences<\/h2>\n

    We will now explore recursively defined sequences of complex numbers. Recursively defined sequences are sequences in which subsequent terms are constructed based on preceding terms using a specific set of rules or formulas, allowing each term to be defined as a function of its predecessors.<\/p>\n

    \n
    \n

    Recursive Sequence<\/h3>\n

    A recursive relationship<\/strong> is a formula which relates the next value, [latex]{{z}_{n+1}}[\/latex], in a sequence to the previous value, [latex]{{z}_{n}}[\/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[\/latex].<\/p>\n

     <\/p>\n

    The sequence of values produced is the recursive sequence<\/strong>.<\/p>\n<\/div>\n<\/section>\n

    In mathematics, we often use subscripts to denote specific elements or terms in a sequence or set. Imagine you have a list of data entries represented by the symbol [latex]x[\/latex]. These entries can be labeled as [latex]x_1,x_2,x_3, ...[\/latex], where the subscript indicates the position of the entry in the list.<\/p>\n

    For instance, [latex]x_1[\/latex] refers to the first entry, and [latex]x_2[\/latex] refers to the second entry, and so on. We sometimes start our list with a “zeroth” term, denoted as [latex]x_0[\/latex]. When we want to talk about a general term in the list, we use [latex]x_n[\/latex], where [latex]n[\/latex] represents any position in the list. This notation also allows us to refer to the terms before and after [latex]x_n[\/latex] as [latex]x_{n-1}[\/latex] and [latex]x_{n+1}[\/latex], respectively.<\/p>\n<\/section>\n

    \n

    How To: Apply a Recursive Sequence<\/strong><\/p>\n

      \n
    1. Identify the Initial Term:<\/strong> Start by determining the first term of the sequence, often denoted as [latex]a_0[\/latex] or [latex]a_1[\/latex], provided in the sequence definition. This term serves as the starting point for building the rest of the sequence.<\/li>\n
    2. Understand the Recursive Formula:<\/strong> Look at the recursive relationship that defines how to find each term from the previous term(s). The formula will generally be given in the form of [latex]a_{n+1}=f(a_n)[\/latex], where [latex]f[\/latex]\u00a0represents a function.<\/li>\n
    3. Apply the Formula:<\/strong> Use the formula to calculate the next term in the sequence by substituting the previous term into the formula.<\/li>\n
    4. Repeat the Process:<\/strong> Continue applying the recursive formula to each new term to find subsequent terms as needed.<\/li>\n<\/ol>\n<\/section>\n
      Given the recursive relationship [latex]{{z}_{n+1}}={{z}_{n}}+2, {{z}_{0}}=4[\/latex], generate several terms of the recursive sequence.<\/p>\n