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MIT6_042JF10_rec04

MIT6_042JF10_rec04 - numbers F n and F n 1 are relatively...

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6.042/18.062J Mathematics for Computer Science September 22, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 4 1 Problem: The Pulverizer! There is a pond. Inside the pond there are n pebbles, arranged in a cycle. A frog is sitting on one of the pebbles. Whenever he jumps, he lands exactly k pebbles away in the clockwise direction, where 0 < k < n . The frog’s meal, a delicious worm, lies on the pebble right next to his, in the clockwise direction. (a) Describe a situation where the frog can’t reach the worm. (b) In a situation where the frog can actually reach the worm, explain how to use the Pulverizer to find how many jumps the frog will need. (c) Compute the number of jumps if n = 50 and k = 21. Anything strange? Can you fix it?
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Recitation 4 2 2 Problem: The Fibonacci numbers. The Fibonacci numbers are defined as follows: F 0 = 0 F 1 = 1 F n = F n 1 + F n 2 (for n 2) . Give an inductive proof that the Fibonacci
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Unformatted text preview: numbers F n and F n +1 are relatively prime ±or all n ≥ 0. ±² ³ Recitation 4 3 3 Extra Problem: The power of 3. 1 Let N be a number whose decimal expansion consists of 3 n identical digits. Show by induction that 3 n N . For example: | 3 2 | 777777777 3 2 = 9 digits Recall that 3 divides a number iff it divides the sum of its digits. 1 Try this if you have time! MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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