Fall 2003 Examination 2 CMSC 203
1. Circle T if the corresponding statement is True or F if it is False.
TFStarting from 1, the sum of the first 200 integers is 10,100.
TFThe First (Weak) and Second (Strong) Principles of Mathematical Induction can be
used interchangably.
TFAlgorithms of constant order take the same amount of time to execute, regardless of
the inputs.
TFAny two distinct prime numbers have 1 for a Greatest Common Divisor.
TFFunctions that are O(x1000) grow faster than functions that are O(2x).
TFFor all integers a, b, and c, if a divides (b + c), then a divides b and a divides c.
TFThe product of an Irrational and an Irrational is Irrational.
TFFor positive integers, n, GCD(n, 0) = n.
2. Given the recursion defined as f(n) = f(n
โ
1) + f(n
โ
2), find the next 4 terms when:
(a) f(0) = 2 and f(1) = 4 (b) f(0) =
โ
10 and f(1) = 5
3. Let {an} and {bn} be the sequences defined, for n > 0, by: an = 3n , bn = 2n.
Find c0, c1, c2, and c3 when cn = an
โ
bn.
4. (a) Find the Big-Oh of the algorithm with complexity:
(n5 + 1)(n + n2) + (5n2 + 3n + 2)(n5).
(b) The following algorithms evaluate the polynomial anxn + an โ 1xn โ 1 +...+ a1x + a0 at x = c.
Step through each to evaluate x3 + x2 + x + 1 at x = 2.
procedure poly1 (c, a0, a1, ..., an : real numbers)
power := 1
y := a0
for i = 1 to n
begin
power := power * c
y := y + ai * power
end
output y
procedure poly2 (c, a0, a1, ..., an : real numbers)
y := an
for i = 1 to n
begin
y := y*c + an โ i
end
output y
5. Use the Euclidean Algorithm to find GCD(1000,256).
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