CMSC 203 - Fall 2005 - Exam 2
1. Circle T if the corresponding statement is True or F if it is False.
TFThe sequence {1, 1, 2, 3, 5, 8, 13, 21,...} satisfies the Fibonacci relation.
TFThe First (Weak) Principle of Mathematical Induction implies the Second (Strong)
Principles of Mathematical Induction.
TFA recursive algorithm is one which never terminates.
TF2 + 4 + 6 + 8 + ... + 200 = 10100.
TFFunctions that are Logarithmic order grow faster than functions that are Linear order.
TFIn costing algorithms, the sum of the orders equals the maximum order.
TFIn costing algorithms, the product of the orders equals the order of the product.
TFIn general, Linear Search algorithms are just as fast Binary Search algorithms.
2. Let {an} and {bn} be the sequences defined, for n > 0, by:
an = n2 − 2, and bn = n3 + 2. Find c0, c1, c2, and c3 when cn = (an)(bn).
3. Given positive integer inputs, A and B, write out in pseudocode the algorithm to
calculate (A MOD B) and (A DIV B). You may assume B < A.
4. Rank from 1 (least complex) to 10 (most complex) the complexity of algorithms
with the following orders:
Order: 2nlognn! n21nn
nnlognn10 10n
Rank:
(b) Find the Big-Oh of the algorithm with complexity: (n4 + 1)(2n + n2) + (3n2 + 4n4 + 5)(n2).
5. Use the Euclidean Algorithm to find GCD(1100, 35).
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 = 0 and a1 = 10, then an = an−1 + an−2 is divisible by 10.
7. Prove one of the two Theorems below:
Theorem 1: If an integer divides two other integers, then it divides their sum.
Theorem 2: If a and b are odd, then (a2 + b2 ) is even.
8. Prove one of the two Theorems below by Contradiction.
Theorem 1: If a prime divides an integer, then it cannot divide the integer’s successor.
Theorem 2: There does not exist a largest integer.
ai
i0=
n
∑an1+ 1–
a1–
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