## Another question about iterating weighted serie to convergence

I have weighted function

$a_1 = a_0(1-\alpha) + b\alpha$

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"Running-Gag" solved the my previous question:

You can show that
Therefore to get for some given you must have
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I have many source $a$ and destination $b$ values i.e. signal but I have to use only one $\alpha$.

If target is defined such that $p=\frac{b-a_n}{b-a_0}$
Then iterating I can find e.g that if
$\alpha=\frac{2}{n}$
$\lim_{n\rightarrow\inf}p=0.135$

So what is function for $\alpha$ if $p$ and $n$ is given ?:
$\alpha=f(p,n)$