# A first order difference equation

• Mar 5th 2010, 08:58 PM
FoxyGrandma3000
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:

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

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

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

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

but I'm completely lost on the jump between the two. If anyone could shed some light on this it would be much appreciated.
• Mar 5th 2010, 09:05 PM
TheEmptySet
Quote:

Originally Posted by FoxyGrandma3000
I'm reading a chapter on first order difference equations in which they use the example of population growth given by the model:

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

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

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

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

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

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

Geometric series - Wikipedia, the free encyclopedia
• Mar 5th 2010, 09:09 PM
FoxyGrandma3000
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.