Summarize my formula?

Question

I would like to summarize my formula. $p$ and $y$ are constant value, $10000$ and $0.65$.

When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to summarize or other formula for that equivalent?

$$x_n=(p+x_{n-1})y$$

Update :

For More Info :

$$\begin{align}n=1 &\Rightarrow 10000 * 0.65 = 6500\\ n=2 &\Rightarrow (10000 + 6500) * 0.65 = 10725\\ n=3 &\Rightarrow (10000 + 10725) * 0.65 = 13471.25\end{align}$$

If I put $n=7$, I don't want to recalculate the result of $n=1$ to $n=6$.

Answer 1

so we have like this yes

$x_n=(10000+x_{n-1})*0.65$

so we have

$x_n=6500+0.65*x_{n-1}$

rewrite it as

$x_n-0.65*x_{n-1}=6500$

http://www.economics.utoronto.ca/osborne/MathTutorialSF/FODF.HTM

see this please,important point is to see pattern,like try to use recurrent relationship like this

$y_n=3*y_{n-1}$ for example $y_{n-1}=3*y_{n-2}$ now put what is $y_{n-1}$ from the previous,it is like this

Answer 2

Already answered at http://www.stackoverflow.com/questions/20465190/summarize-my-formula. The general tool to deal with difference equations is Z-Transformation. The solution for the wanted initial condition $x_0=0$ is \begin{align*} x_n = py\frac{1-y^n}{1-y}. \end{align*}