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Math Help - A first order difference equation

  1. #1
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    A first order difference equation

    I'm reading a chapter on first order difference equations in which they use the example of population growth given by the model:

     y_{n}=\rho^{n}y_{0}+(1+\rho+\rho^{2}+\cdot\cdot\cd  ot+\rho^{n-1})b

    Where  \rho is the reproduction rate and  b is the rate of immigration.

    They then go on to assume that if  \rho\neq1 they can simplify the equation to the compact form:

     y_{n}=\rho^{n}y_{0}+\frac{1-\rho^{n}}{1-\rho}b

    but I'm completely lost on the jump between the two. If anyone could shed some light on this it would be much appreciated.
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  2. #2
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    Quote Originally Posted by FoxyGrandma3000 View Post
    I'm reading a chapter on first order difference equations in which they use the example of population growth given by the model:

     y_{n}=\rho^{n}y_{0}+(1+\rho+\rho^{2}+\cdot\cdot\cd  ot+\rho^{n-1})b

    Where  \rho is the reproduction rate and  b is the rate of immigration.

    They then go on to assume that if  \rho\neq1 they can simplify the equation to the compact form:

     y_{n}=\rho^{n}y_{0}+\frac{1-\rho^{n}}{1-\rho}b

    but I'm completely lost on the jump between the two. If anyone could shed some light on this it would be much appreciated.
    It is the sum of a geometric series

    \sum_{n=0}^{k}r^{n}=\frac{1-r^{k+1}}{1-r}, |r|<1

    Geometric series - Wikipedia, the free encyclopedia
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  3. #3
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    Thanks...I actually figured it out right after I posted it and felt really dumb.

    I guess that's what you get for not doing any math for 5 years and then cracking open a diff eq book.
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