CMPSCI 190DM: A Mathematical Foundation for Informatics

Practice for First Second Exam

David Mix Barrington

17 October 2015

Directions:

Answer the problems on the exam pages.
There are four problems for 100 total points. Scale will be determined after the exam but a good guess is A = 90, C = 60.
If you need extra space use the back of a page.
No books, notes, calculators, or collaboration.

  Q1: 15 points
  Q2: 15 points
  Q3: 20 points
  Q4: 20 points
  Q5: 30 points
Total: 100 points

Corrections in green made on 20 October 2015.

Question 1 (15): Briefly identify the following terms or concepts (3 points each):

(a) for x to be a multiple of y
(b) a proof by contradiction
(c) a rational number
(d) the base case of a proof by induction
(e) a predicate on the positive integers

Question 2 (15): Determine whether each of these five statements is true or false. No explanation is needed or wanted, and there is no penalty for guessing.

(a) 91 is not divisible by 7.
(b) (31 mod 6) = (25 mod 4) (Remember that E&C's "mod" is the same as Java's "%" on non-negative integers.)
(c) The binary representation of the decimal number "19" is "10101".
(d) Every perfect square can be written as either "4k" or "4k+1" for some integer k.
(e) The square of every irrational number is irrational.

Question 3 (20): Prove that if k is any integer, the number 7k + 5 is not a perfect square. (Hint: Argue the contrapositive, and apply the Division Theorem with m = 7 to the number you are squaring.)

Question 4 (20): Define a number sequence so that a₁ = 3 and for all k with k > 1, a_k = a_k-1 + 4k - 7. Prove by induction that for any positive integer n, a_n = 2n² - 5n + 6.

Question 5 (30): Define a number sequence so that b₁ = 2, b₂ = 3, and for all k with k > 2, b_k = b_k-1 - b_k-2 + 1.

(a, 5) Calculate b_i for all integers i from i through 10.
(b, 5) Determine the value of b₁₀₀. (Hint: I don't expect you to determine all the intermediate values -- find a pattern in part (a) or use the result of part (c).)
(c, 20) Prove by induction that for any n with n > 6, b_n = b_n-6. (Hint: Prove the base cases of n = 7 and n = 8, then use the sequence definition in your inductive case.)

Last modified 20 October 2015