Fall 2007 - CMSC203 - Discrete Structures - Exam 2
1. Circle T if the corresponding statement is True or F if it is False.
TFThe sequence {0, 1, 4, 9, 16, ...} is an example of an Geometric sequence.
TFThe Principle of Mathematical Induction can be used to create new formulas.
TFAny recursive algorithm has an equivalent iterative version.
TF10 + 20 + 30 + 40 + ... + 10000 = 5005000.
TF2 + 2 + 4 + 8 + 16 + ... + 219 = 220.
TFAlgorithms whose order is O(log n) are less efficient than those of O(n).
TFThe Euclidean Algorithm has exponential order.
TFLinear Search algorithms are as efficient as Binary Search algorithms.
2.Let {an} and {bn} be the sequences defined, for n > 0, by:
an = (โ1)(n + 1), and bn = (n2+ 1). Find c0, c1, c2, and c3 when cn = (an)(bn).
3. Trace the steps of the Division Algorithm to calculate (21 MOD 4) and (21 DIV 4).
4. (a) Give a Recursive Definition for the Natural Numbers.
(b) Find the Big-Oh of the algorithm with complexity: (n + 4)(3n2 + 1) + (n2 + 2n + 2)(n2).
5. (a) Given a = 253250771141381711194237293318.
and b = 28365775110139172191223529133111, then
GCD(a,b) = ___________________ and LCM(a,b) = ___________________.
(b) Use the Euclidean Algorithm to find GCD(200, 44).
6. Prove one of the two Theorems below using Mathematical Induction.
Theorem 1: For all integers n > 1, .
Theorem 2: If a0 = 0, a1 = 50, and a2 = 100, then an = anโ1 + anโ2 + anโ3 is divisible by 50.
7. Prove one of the two Theorems below:
Theorem 1: Given that either a Natural Number or its successor is even, if n is an odd Natural
Number, then (n2 MOD 8) = 1.
Theorem 2: If a, b, q are Integers with q > 1, and (a MOD q) = (b MOD q), then (a โ b) is a
multiple of q.
8. Prove one of the two Theorems below by Contradiction or Contraposition.
Theorem 1: If n is a Natural Number and n2 is even, then n is even.
Theorem 2: The Natural Numbers are an infinite set.
i2
i1=
n
โnn 1+()2n1+()
6
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