SAMPLE Exam 2 - Fall 2006 - Discrete Structures
1. Circle T if the corresponding statement is True or F if it is False.
TFThe sequence {1, โ1, 1, โ1, 1, โ1,...} is an example of an alternating sequence.
TFThe Principle of Mathematical Induction can be used to discover new theorems.
TFA recursive algorithm is generally more efficient in memory than its equivalent
iterative version.
TF3 + 6 + 9 + 12 + ... + 300 = 15150.
TFAlgorithms that are Exponential order grow faster than ones that are Polynomial order.
TFIn costing algorithms, the sum of the orders equals the maximum order.
TFThe Fibonacci sequence only requires one initial condition.
TFIn general, Linear Search algorithms are less efficient than Binary Search algorithms.
2. Let {an} and {bn} be the sequences defined, for n > 0, by:
an = n โ 2, and bn = 3n + 1. 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. (a) Rank from 1 (least complex) to 5 (most complex) the following orders:
___ 10n ___ n2___ n ___ nn___ n10
(b) Find the Big-Oh of the algorithm with complexity: (n6 + 1)(2n2 + 3n) + (n3 + 5n2 + 2)(n3).
5. Use the Euclidean Algorithm to find GCD(100, 22).
6. Prove one of the two Theorems below using Mathematical Induction.
Theorem 1: For all integers n > 0 and a โ 0,1, .
Theorem 2: If a0 = 0 and a1 = 100, then an = anโ1 + anโ2 is divisible by 100.
7. Prove one of the two Theorems below:
Theorem 1: The product of odd integers is odd.
Theorem 2: If (a MOD p) = (b MOD p), then p divides (a โ b).
8. Prove one of the two Theorems below by Contradiction.
Theorem 1: is irrational.
Theorem 2: There does not exist a largest integer.
ai
i0=
n
โan1+ 1โ
a1โ
-------------------=
2
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