Module 9 Homework

1. Let \(\{a_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(a_1=4\) and recurrence relation \(a_n=\frac{1}{a_{n-1}}\). Write out the first eight terms of this sequence.
2. Let \(\{b_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(b_1=1,\ b_2=2\ b_3=4\) and recurrence relation \(b_n=b_{n-1}+2\cdot b_{n-2}+b_{n-3}\). Write out the first eight terms of this sequence.
3. Let \(\{d_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(d_1=1,\ d_2=2\ d_3=4\) and recurrence relation \(d_n=\frac{d_{n-1}\cdot 2\cdot d_{n-2}}{d_{n-3}}\). Write out the first seven terms of this sequence.
4. Let \(\{d_n\}_{n=1}^\infty\) be a recursively defined sequence of images with initial condition and recurrence relation given below. Generate the first four elements of this sequence. (What the given images are saying is “you start with a line, and from there, you generate subsequent elements of the sequence by replacing the middle third of all straight lines you see with a little triangle.”) The image you get is a fractal called the “Koch Snowflake.” Note that all fractals are generated by recurrence relations.

5. Let \(\{T_n\}_{n=1}^\infty\) be a recursively defined sequence of images with initial condition and recurrence relation given below. Generate the first six elements of this sequence. (What the given images are saying is “you start with a line, and from there, you generate subsequent elements of the sequence by drawing two smaller lines out the top of any line present so far.”)

6. (Bonus +5) Let \(C=\{-1,1-1,1,-1,1,…\}\). Find a recurrence relation that generates this sequence. That is, given that the initial value of the sequence is \(c_1=-1\) (because it’s the first entry of the sequence), find a formula for \(c_n\) that involves at least one of \(c_{n-1}, c_{n-2}, c_{n-3}, \) etc. so that your recursive formula gives you \(c_2=1,\ c_3=-1,\ c_4=1, \) etc.

7. Let \(\{a_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(a_1=2\) and recurrence relation \(a_n=7+a_{n-1}\). Guess a closed formula for this sequence (be sure to show or somehow indicate how you arrived at your answer)
8. Let \(\{b_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(b_1=2\) and recurrence relation \(b_n=7\cdot b_{n-1}\). Guess a closed formula for this sequence (be sure to show or somehow indicate how you arrived at your answer)
9. Let \(\{v_n\}_{n=1}^\infty\) be a recursively defined sequence with initial conditions \(v_1=1\) and recurrence relation \(v_n=v_{n-1}+2^n\). Guess a closed formula for this sequence (be sure to show or somehow indicate how you arrived at your answer). If your answer uses sigma notation, this is fine.

10. Let \(\{a_n\}_{n=0}^\infty\) be a recursively defined sequence with initial conditions \(a_0=4\) and recurrence relation \(a_n=9+a_{n-1}\). Prove that \(a_n=4+9n\) for all \(n\geq 0\).
11. Prove that the closed formula you got for problem 8 is correct. If it is not correct, then bad/impossible things may happen in the arithmetic you use in your induction argument.
12. Prove that the closed formula you got for problem 9 is correct.

13. (50 points) Suppose you have $500 to invest in Bitcoin (BTC) (yes, you can buy fractions of a bitcoin), and you know that Bitcoin has an average return of, say 90% per year on average (i.e. for every year you hold BTC, your money gets multiplied by 1.90 on average (assuming you don’t sell it for years). Believe it or not, this rate is vastly less than how BTC actually performed over the last 10 years).
    1. Determine how much money you will have after 1, 2, 3, 4, and 5 years.
    2. Find a recursively defined formula based on your answer to part 1 above (i.e. determine an initial condition and a recurrence relation that tells you how much you will have after n years)
    3. Using your recurrence relation, guess a closed formula that tells you how much money you will have after n years.
    4. Prove that this formula is correct.
    5. Determine how much money you will have after 10 years.

(Note that if you ever want to calculate the return on investment (ROI) based on an annual rate of return and annual compounding based on a different starting amount or different rate, you probably can see how to change up your answer to numbers 3 through 5 to make that happen)