CMPSCI 190DM: A Mathematical Foundation for Informatics

Second Midterm Exam

David Mix Barrington

23 October 2015

Directions:

Answer the problems on the exam pages.
There are four problems for 100 total points. Actual scale was A = 90, C = 60.
If you need extra space use the back of a page.
No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 15 points
  Q3: 20 points
  Q4: 20 points
  Q5: 25 points
Total: 100 points

Question 1 (20): Briefly identify the following terms or concepts (4 points each):

(a) the Pigeonhole Principle
(b) an odd number
(c) the Division Theorem
(d) the mod operation
(e) a perfect square

Question 2 (15): Identify each of the following statements as true or false. No explanation is needed or wanted, and there is no penalty for guessing. (3 points each)

(a) The binary representation of the decimal number 36 is 100010.
(b) If S is a set of eleven different positive integers, there exist two elements x and y of S, with x ≠ y, such that x and y have the same last digit in decimal notation.
(c) The sum of any two rational numbers is rational.
(d) Every number that is divisible by 6 is also divisible by 12.
(e) If n is an odd number, then (n-3)³ + 6 must be an even number.

Question 3 (20): Prove that if an integer n is not divisible by 4, then n³ is also not divisible by 4. (Hint: Use Proof by Cases and the Division Theorem on n, with m = 4.)

Question 4 (20): Define a number sequence so that a₁ = 3 and, for any n with n > 1, a_n = a_n-1 + 3n² - 3n + 1. Prove by induction that for all positive integers n, a_n = n³ + 2.

Question 5 (25): The Tribonacci sequence is defined by the rules T₁ = 1, T₂ = 1, T₃ = 1, and for all n with n > 3, T_n = T_n-1 + T_n-2 + T_n-3.

(a, 5) Compute T_k for all k such that 1 ≤ k ≤ 7.
(b, 20) Prove by induction that for all positive integers n, T_n is odd. You will need separate base cases for n = 1, n = 2, and n = 3.

Last modified 25 October 2015