math 302 EC2

math 302 EC2 - 4. Let ( x i ,y i ), i = 1 , 2 , 3 , 4 , 5...

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MATH 302 Discrete Mathematics Extra-credit Assignment 2. Please show your argument and computation. Calculators and computers are not permitted. 1. Let f n be the n -th Fibonacci number. Prove that f 0 f 1 + f 1 f 2 + ··· + f 2 n - 1 f 2 n = f 2 2 n when n is a positive integer. 2. Find the solution to the recurrence relation a n = a n - 1 + 2 a n - 2 with a 0 = 2 and a 1 = 7. 3. In how many ways can one choose 10 integers a 1 ,a 2 ,...,a 10 from the range [1 , 100] such that for any pair of the chosen numbers, the difference is at least 2?
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Unformatted text preview: 4. Let ( x i ,y i ), i = 1 , 2 , 3 , 4 , 5 be a set of ve distinct points with integer coordinates in the xy plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates. 5. For n Z + , consider the following sum n X i =1 1 (2 i-1)(2 i + 1) . make a conjecture about the formula for the exact value of this sum, and prove your conjecture....
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