# Second Midterm Exam

### 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, an = an-1 + 3n2 - 3n + 1. Prove by induction that for all positive integers n, an = 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, Tn = Tn-1 + Tn-2 + Tn-3.

• (a, 5) Compute Tk for all k such that 1 ≤ k ≤ 7.

• (b, 20) Prove by induction that for all positive integers n, Tn is odd. You will need separate base cases for n = 1, n = 2, and n = 3.