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Chapter 1 Sequences and Recursion

Chapter 1 Sequences and Recursion. When might you see recursion in your life?. What is recursion?. Recursion is a process in which each step of a pattern is dependent on the step or steps that come before it.

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Chapter 1 Sequences and Recursion

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  1. Chapter 1 Sequences and Recursion When might you see recursion in your life?

  2. What is recursion? Recursion is a process in which each step of a pattern is dependent on the step or steps that come before it. Recursion occurs when a pattern is repeated an infinite amount of times. The Santa Maria Novella cathedral in Florence, Italy shows a repeated pattern.

  3. Nature Trees are recursive in nature. A tree has branches and each branch has sub branches and so on.

  4. Fibonacci Sequence • There is a fascinating connection between the Fibonacci series of numbers and various natural occurrences. • Starting with one pair of baby rabbits, in one month they become fertile and can mate. After one more month gestation, another pair of rabbits are born. So after those initial 2 months there exists 2 pairs (equals 4 rabbits). Assuming the rabbits live in optimum conditions and that they never die, Fibonacci’s numbers model how the population will grow.

  5. Fractals A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called self-similarity. Sierpinski Triangle

  6. Movie Backgrounds Often in movies, backgrounds are designed by the idea of recursion. Mountain range

  7. Computer Programming Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. Most computer programming languages support recursion by allowing a function to call itself within the program text.

  8. Loans and Investments Recursion is used when investing money as in a Certificate of Deposit and also when paying off loans as in buying a car. Suppose you are borrowing $22,000. Interest is being charged at an annual rate of 5%. You plan to pay back $250 each month; this payment goes to both interest and principal. un=(1+0.0512)un−1−250, for n≥1 You are saving for the future. Your initial deposit is $4100. Interest is being earned at an annual rate of 5%, compounded monthly. You will contribute an additional $120 each month. un=(1+0.0512)un−1+120, for n≥1

  9. Recursion… is the basis of patterns which become the basis of mathematical (algebraic) modeling. It is used all around us and everywhere in the world.

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