Discrete Mathematics with Graph Theory (3rd Edition) 142

Discrete Mathematics with Graph Theory (3rd Edition) 142 -...

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140 Solutions to Exercises any sequence adding to n or to n - 1 can be turned into a sequence adding to n + 1 by adding a 1 or a 2 (respectively) at the end. Thus, bn+1 = bn + bn- I. Adding the initial conditions b l = 1, b 2 = 2 gives the desired recursive definition and proves that the sequence is the Fibonacci sequence, starting with the second tenn. 55. (a) [BB] Each line of n - 1 people determines two lines of n people-the first person in the new line can be a man or a woman. So an = 2an -l. The sequence is the sequence 2, 4, 8, ... of powers of2. (b) A line starts with a male or a female. If it starts with a female, there are an-I ways to complete the line. On the other hand, if it starts with a male, then the second person must be female. There are an-2 ways then to complete such a line. Thus, an = an-I + an-2. Since al = 2 (M or F) and a2 = 3 (FF, MF, or PM), the next few tenns are 5, 8, 13, . ... This is that part of the Fibonacci sequence from 2 onward. 56. (a) We use induction on
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