sol10 - AMS 301 (Spring 2011) Homework Set # 10 Solution...

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Unformatted text preview: AMS 301 (Spring 2011) Homework Set # 10 Solution Notes # 6, 7.1: (a). a n = a n 1 + a n 2 . If the first digit is a 0, there are a n 1 ways to complete the sequence. If the first digit is a 1, then the second digit must be a 0, and then there are n- 2 digits left, so a n 2 ways to complete the sequence. a 1 = 2, a 2 = 3 are the initial condition. (b). a n = 2 a n 1 + 2 a n 2 . If the first digit is a 0, or a 2, there are a n 1 ways to complete the sequence. If the first digit is a 1, then the second digit must be a 0, or 2, and then there are n- 2 digits left, so a n 2 ways to complete the sequence. a 1 = 3, a 2 = 8 are the initial condition. # 24, 7.1: In this case once a digit 2 appears, all digits to its right must be 0 and 2s (no 1s allowed). Our recursion conditions on the first digit. If it is a 0 or 1 (two choices) then any sequence of length n- 1 that has no 1 to the right of a 2 can follow. However, if the first digit is a 2, the remaining1 that has no 1 to the right of a 2 can follow....
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