challenge1 - F 1 = 1 F 2 = 1 and for n ≥ 3 F n = F n-1 F...

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MATH 135, Challenge Problem Set 1 Warning: These problems are not intended as additional practice problems to help you prepare for tests. They are more difficult (usually considerably more difficult) than the assigned homework problems, and they are intended for students who would like to work on some more challenging problems. Do not be upset if any (or all) of them stump you; mathematicians routinely wrestle with problems that defeat them. 1: Determine whether there exists a positive integer n such that n 2 + n +1 is a perfect square. 2: Let a 0 = c and for n 1 let a n = pa n - 1 + q where p = 0. Find a (non-recursive) formula for a n in terms of c , p and q . 3: Let { F n } be the Fibonacci sequence (so
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Unformatted text preview: F 1 = 1, F 2 = 1, and for n ≥ 3, F n = F n-1 + F n-2 ). Prove that F n 2 + F n +1 2 = F 2 n +1 for all n ≥ 1. 4: Find the number of ways to roll six identical 6-sided dice (one such way is to roll two 3s, three 5s and one 6). 5: Let a = 9 and for n ≥ 0 let a n +1 = 3 a n 4 + 4 a n 3 . Prove that for all n ≥ 0, at least 2 n of the digits in a n are nines. 6: There are n points on a circle. Each pair of points is connected by a line segment. No three of these line segments meet at a point. Find the number of regions into which the circle is divided....
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