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.