Counting Bug PopulationsDate: 12/03/98 at 09:04:16 From: Naveed Ahmed Subject: Probability, sequences, formulas We are given the following information: A happy bug splits into a sad bug and a blank bug. A sad bug splits into 2 happy bugs. A blank bug splits into a sad bug and a happy bug. The generation starts off with 1 happy bug which splits and forms the 2nd generation. The bugs die when they split. The bugs live the same amount of time. I have worked the first five generations to find how many of each kind of bugs there are, but I can't find the formula for the nth generation. Could also you tell me the formulas if the first generation starts with a sad bug and a blank bug? Thanks. Date: 07/26/2017 at 11:05:08 From: James Subject: Re: Probability, sequences, formulas Let: n = generation H = happy bug S = sad bug B = blank bug Using the given information, the table should be: n 1 2 3 4 5 6 7 8 9 10 11 12 ---------------------------------------------------------------- H 1 0 3 2 9 12 31 54 117 224 459 906 B 0 1 0 3 2 9 12 31 54 117 224 459 S 0 1 1 3 5 11 21 43 85 171 341 683 ----------------------------------------------------------------- Total 1 2 2^2 2^3 2^4 2^5 2^6 2^7 2^8 2^9 2^10 2^11 Assuming n is a whole number starting at zero, this would allow the equations to be: h(n) = 1/9*(3*(-1)^n*n + 5*(-1)^n + 2^(n + 2)) s(n) = 1/3*(2^n - (-1)^n) b(n) = (1/9)*((-1)^(n + 1)*(3*n + 2)+2^(n + 1)) t(n) = 2^n Date: 07/27/2017 at 11:57:13 From: Doctor Peterson Subject: Re: Counting Bug Populations Hi, James. I agree, the correct recursion is H --> B + 2S B --> H S --> H + B You didn't say how you obtained your formulas, so I can't definitely say they are correct, but they appear reasonable. - Doctor Peterson, The Math Forum at NCTM http://mathforum.org/dr.math/ Date: 07/27/2017 at 12:03:54 From: James Subject: Thank you (Counting Bug Populations) Thank you. I wasn't able to figure out how to generate the equations myself, so I utilized Wolfram|Alpha's equation generator to derive h(N). The other equations naturally came after. |
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