Homework Set 2

Submitting Your Work

Your work must be submitted to Anchor for degree credit and to Gradescope for grading.

For any work completed outside of GitHub or Gradescope:

  1. Typeset your work in LaTeX.
  • You should use \begin{enumerate} to create a numbered list, and your solutions to each problem should match with the corresponding number on the assignment.
  • For example, if an assignment has 10 numbered problems, your enumerate should have 10 \item commands, with your solution(s) to problem 1 under the first \item, problem 2 under the second and so on.
  • If you skip a problem, just leave the \item blank, otherwise the numbers of your solutions will not match the assignment document. Here is an example Overleaf document showcasing this format.
  1. Compile into a PDF. This can be done easily through an Overleaf account, otherwise you will need to install LaTeX.
  2. Submit the pdf to Gradescope via the appropriate submission link for the course.
  3. Upload the pdf to Anchor using the form below.

Note: Anchor submissions can occur at any time during the term, but it is critical that you upload all of your work to Anchor before the last day of the term. Gradescope submissions must be submitted before the deadline (or the late deadline, if applicable).

It is required that all assignments are submitted as a PDF generated from a LaTeX document.

Assignments submitted in any other form will earn zero credit.

  1. [35 points] Find the closed form for the following series, up to the $n$th term. If the infinite series has a finite solution, write it down.

    a. 1 + 4 + 7 + 10 + 13 + ...

    b. $a_n = 3 \cdot a_{n-1}$, $a_0 = 2$

    c. 5 + 17 + 29 + 41 + 53 + ...

    d. $a_n = a_{n-1} + 4$, $a_0 = -1$

    e. $\frac{1}{2} + \frac{1}{6} + \frac{1}{18} + \frac{1}{54} + \frac{1}{162} + ... $

    f. -15 + -13 + -11 + -9 + -7 + ...

    g. $\frac{1}{5} - \frac{1}{35} + \frac{1}{245} - \frac{1}{1715} + \frac{1}{12005} - ...$

  2. [5 points] Is the Fibonacci sequence an arithmetic sequence? Explain your answer.

  3. [5 points] Is the sequence $a_n = 2^n$ a geometric sequence? Explain your answer.

  4. [5 points] Which of the following is true:

    • Functions are a subset of sequences.
    • Sequences are a subset of functions.
    • Functions are a subset of relations.
    • Relations are a subset of functions.
    • None of these are true.

  5. [10 points] Are all linear functions bijective? Prove or disprove.

  6. [10 points] Are all quadratic functions bijective? Prove or disprove.

  7. [10 points] Are all polynomial functions bijective? Prove or disprove.

  8. [10 points] Prove that the set of all odd integers is countable.