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f35-frosh-sem-easy-hard-handout

# f35-frosh-sem-easy-hard-handout - Easy Euclid Sequences...

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Mar. 2011 Easy, Hard, Impossible! Handout Easy: Euclid Sequences Form a sequence of number pairs (integers) as follows: Begin with any two positive numbers as the first pair In each step, the next number pair consists of (1) the smaller of the current pair of values, and (2) their difference Stop when the two numbers in the pair become equal Challenge: Repeat this process for a few more starting number pairs and see if you can discover something about how the final number pair is related to the starting values (10, 15) (10, 5) (5, 5) (9, 23) (9, 14) (9, 5) (5, 4) (4, 1) (1, 3) (1, 2) (1, 1) (22, 6) (6, 16) (6, 10) (6, 4) (4, 2) (2, 2) Why is the process outlined above guaranteed to end?

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Mar. 2011 Easy, Hard, Impossible! Handout Not So Easy: Fibonacci Sequences Form a sequence of numbers (integers) as follows: Begin with any two numbers as the first two elements In each step, the next number is the sum of the last two numbers already in the sequence Stop when you have generated the j th number ( j is given) Challenge: See if you can find a formula that yields the j th number directly (i.e., without following the sequence) when we begin with
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