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Understanding Fibonacci Sequence and Iteration Methods in Mathematics

Dive into the fascinating world of the Fibonacci sequence and iteration techniques in mathematics. Explore how the Fibonacci sequence develops through the pattern of rabbit reproduction, where each rectangle symbolizes a rabbit pair. Learn how to iteratively compute function outputs using previous values—key for understanding sequences. We’ll walk through examples to illustrate the first four iterations of specific functions, rounding to the nearest hundredth. Discover the power of sequences, iterations, and their applications in problem-solving.

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Understanding Fibonacci Sequence and Iteration Methods in Mathematics

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  1. 12.7 – Fibonacci Sequence

  2. I can recognize the Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … What’s the pattern?

  3. Rabbits and Fibonacci • Each small rectangle will represent a pair of baby rabbits. • It takes one month for a baby rabbit to mature and reproduce. (Represented by a big square tile) • Assume each mature rabbit reproduces one pair of baby rabbits every month and never die.

  4. 12.8 – Sequences & Iteration • I can iterate functions: • Iteration: composing a function with itself. • Iterate: the output of an iteration. • To iterate a sequence you will use the previous term to find the next term.

  5. Find the first four iterates of each function using the given initial value. • f(x) = 3x – 7; x0 = 4 5, 8, 17, 44 3(4) – 7 = 5 3(5) – 7 = 8 3(8) – 7 = 17 3(17) – 7 = 44

  6. Find the first four iterates of each function using the given initial value. Round to the nearest hundreth. 2. f(x) = 2x2 – x; x0 = 0.1 -0.08, 0.09, -0.07, 0.07 2(0.1)2 – 0.1 = -0.08 2(-0.08)2 – -0.08 = 0.09 2(0.09)2 – 0.09 = -0.07 2(-0.07)2 – -0.07 = 0.07

  7. Find the first three iterates of the function f(z) = 2z + (3-2i) for z0 = 1 + 2i 2(1 + 2i) + (3 – 2i) 2 + 4i + 3 – 2i 5 + 2i 2(5 + 2i) + (3 – 2i) 10 + 4i + 3 – 2i 13 + 2i 2(13 + 2i) + (3 – 2i) 26 + 4i + 3 – 2i 29 + 2i

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