The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Counting Bug Populations


Date: 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.
    
Associated Topics:
High School Discrete Mathematics
High School Sequences, Series

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/