Suppose that I want to find a linear function $y(x)=Ax+B$, with coefficients $A$ and $B$ to be determined. The growth of $x$ is governed by the following equation: $x_{i+1}=x_i+qy(x_i)^p$, where $q$ and $p$ are known constants.
Now, given 3 known values of $x$, e.g., $x_0=1$, $x_2=5$ and $x_4=10$, how may I determine $A$ and $B$? This turns out to be a sum of series problem:
$x_1=x_0+qy(x_0)^p=x_0+q(Ax_0+B)^p$
$x_2=x_1+qy(x_1)^p=x_0+q(Ax_0+B)^p + q\{A[x_0+q(Ax_0+B)^p]+B\}^p$
and so on.
Ultimately, this will lead to 2 equations in terms of $A$ and $B$ with the given $x_2$ and $x_4$ values.
Any help or reference to papers will be greatly appreciated. Thank you.