CMSC203 - Spring 2006 - Examination2
1. Circle T if the corresponding statement is True or F if it is False.
TFThe sequence {1, −1, 1, −1, 1, −1,...} is an example of an Oscillating Sequence.
TFA sequence is just another way to describe a function from the Real numbers to
another set.
TFA recursive algorithm can always be expressed equivalently as an iterative algorithm.
TF10100 = 1 + 9(1 + 10 + 100 + ... + 1099).
TFFunctions that are Polynomial order usually grow faster than those that are Exponential order.
TFIn costing algorithms, the sum of the orders equals the order of the largest term.
TFMathematical Induction is useful for finding new summation formulas.
TFIntegers that are Relatively Prime have the same Prime numbers as factors.
2. Let {an} be the sequence defined by: an = n2a(n − 1) . Find a4 when a0 = 2.
3. Fill in the Trace Table of the Division Algorithm when finding (27 DIV 4) and (27 MOD 4).
4. (a) Rank from 1 (least complex) to 10 (most complex) the complexity of algorithms
with the following orders:
Order 10nlog nn! 1 nn
nn10
Rank
(b) Find the Big-Oh of the algorithm with complexity: (n4 + n2)(3n + n4) + (2n3 + n4 + 7)(n3).
5. Use the Euclidean Algorithm to find GCD(99, 21).
6. Prove one of the two Theorems below using Mathematical Induction.
Theorem 1: For all integers n > 0 and a ≠ 0,1, .
Theorem 2: If a0 = a1 = a2 = 10, then an = a(n − 1) + a(n − 2) + a(n − 3) is divisible by 10, for n > 2.
7. Prove one of the two Theorems below:
Theorem 1: If an Integer divides two other Integers, then it divides any linear combination of
the two other Integers. (Definition: For any X and Y, (aX + bY) is a Linear
Combination of X and Y, where a and b are arbitrary constants.)
Theorem 2: If a and b are odd Integers, then (a3 + b3 ) is even.
8. Prove one of the two Theorems below by Contradiction.
Theorem 1: If 3 dividing the square of an Integer implies 3 divides the Integer, then is
irrational .
Theorem 2: No Prime number can divide both an Integer and the Integer’s successor.
ai
i0=
n
∑an1+ 1–
a1–
-------------------=
3
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