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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 Fall '08
 CHEN
 Natural number, Fibonacci number

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