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Practice Quiz 2 (no solutions)

Practice Quiz 2 (no solutions) - n is a natural number and...

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Math 100 Practice Quiz 2 Martin H. Weissman On Friday, November 9, there will be a one-hour quiz in class. The quiz will be very similar, in format and difficulty, to the series of questions given below. 1. (1 point) Let S = { 1 , 3 , 6 } . Let S 2 be the set of ordered pairs, whose entries are elements of S . Let P ( S 2 ) be the set of subsets of S 2 . How many elements does P ( S 2 ) have? (You do not need to simplify your answer). 2. (1 point) How many numbers between 3 and 300 are multiples of 2 or multiples of 3 (or both)? 3. (1 point) Suppose that S is a set with 6 elements. How many subsets of S have 3 elements? 4. (2 points) Prove that n 3 < 2 n , if n
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Unformatted text preview: n is a natural number, and n ≥ 11. To save time, you may use the fact that: n 3 = ( n-1) 3 + 3 n 2-3 n + 1 . 5. (2 points) Let F n be the Fibonacci sequence, given recursively by the following: • F = 1, and F 1 = 1. • If n ∈ N , and n ≥ 2, then F n = F n-1 + F n-2 . Prove that if n ∈ N , and if F n is even, then F n +1 is odd. 6. (3 points) Let S be the set of ordered triples ( x,y,z ), such that x,y,z ∈ Z , x > 0, y > 0, z > 0, and x + y + z = 100. For example, (1 , 9 , 90) ∈ S , and (40 , 40 , 20) ∈ S . How many elements does S have?...
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