Alright, I'm having a really difficult time putting the correct mathematical terms on the problem I'm trying to deal with (which is probably why I can't find an answer), so bear with me.*
0
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*----*----*-------*------------*
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x0 x1 x2 x3
I have a function that linearly increases the distance between successive points based on the position of the previous point.
$$x_{i+1} = x_i + \phi x_i$$
$x_0$ and $x_1$ are given, but $x_0 \neq 0$ . $\phi$ is the linear growth rate
How do I calculate the value of an arbitrary $x_i$ without calculating the value of all the other points?
PS, I'm really not sure what the appropriate tag is because there isn't a "growth" tag. Power series is just a guess as to what the answer might be.
*From the selected answer the sequence described is a geometric sequence
Since $x_{i+1}=(\phi+1) x_i$, you have a geometric sequence:
$$x_1=(\phi+1) x_0$$
$$x_2=(\phi+1) x_1=(\phi+1)(\phi+1) x_0=(\phi+1)^2 x_0$$
In general,
$$x_n=x_0(1+\phi)^n$$