a3-20-08

A - Section 20 5 Show that Fn > 1:6n once n is big enough The basis step here is to nd a place where the inequality is true for two consecutive

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Section 20 5. Show that F n > 1 : 6 n once n is big enough . The basis step for two consecutive values of n . That±s because the induction can±t go from n 1 to n , we generate F n = F n 2 + F n 1 from the two preceding values. Computing F n and 1 : 6 n for lots of values of n F 29 = 832040 > 830767 : 5 = 1 : 6 29 F 30 = 1346269 > 1329228 = 1 : 6 30 The induction step is to show that if F k 2 > 1 : 6 k 2 and F k 1 > 1 : 6 k 1 , then F k > 1 : 6 k . That is, if the inequality is true when n is any number less than k (and, in fact, we need only assume it for n = k 2 and n = k 1 ) then it is true for n = k . In fact F k = F k 2 + F k 1 > 1 : 6 k 2 +1 : 6 k 1 = 1 : 6 k 2 (2 : 6) > 1 : 6 k 2 (1 : 6) 2 = 1 : 6 k So F n > 1 : 6 n for all n ± 29 . Section 21 5. in the line is a man, then somewhere in the line a woman is directly in front of a man . The basis step is to examine the shortest line, which has length two and consists of one woman in front of one man. Clearly the statement is true in this case. The induction step is to pass from the truth of the statement for lines of length k to the truth of the statement for lines of length k +1 . Look at a line of length k + 1 (the front of the line is to the right) M X _ _ ::: _ W If X , the next to the last person in line, is a woman, then there is a woman directly in front of the last person in line, who is a man. If X is a man, then look at the shorter line obtained by removing M , the last man. This is a line of length k that starts with a woman and
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ends with the man X . By the induction hypothesis, our statement is true for lines of the length k . So in that shorter line, there is a woman
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This note was uploaded on 07/13/2011 for the course MAD 2104 taught by Professor Staff during the Spring '08 term at FAU.

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A - Section 20 5 Show that Fn > 1:6n once n is big enough The basis step here is to nd a place where the inequality is true for two consecutive

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