Q:

Calculating the $n$th generation with $p_{n}=\frac{p_{n-1}(1-p_{n-1})}{1-{p_{n-1}}^2}$?

The formula for the next generation $p$ is as following:
$$p_{n}=\frac{p_{n-1}(1-p_{n-1})}{1-{p_{n-1}}^2}$$

$p_n$ is the percentage of the new allele in the next generation
Also $0 <p_{n-1} < 1$

I was wondering if there is just a standard formula for $p_n$, because I was asked to calculate the percentage of the $50$th generation. My only conclusion is that I have to enter the value 50 times into the formula, but I don't know if there is a more efficient way to calculate this.

A:

Assuming $p_{n-1}<1$, a factor of $1-p_{n-1}$ can be pulled out of both the numerator and denominator to get
$$p_n=\frac{p_{n-1}}{1+p_{n-1}} \tag{1}$$
Defining a reciprocal series $q_n=\frac{1}{p_n}$ we get by inverting (1)
$$q_n=q_{n-1}+1 \tag{2}$$
Hence for $k\ge1$ we have by repeated application of (2):
$$q_n=q_{n-k}+k \tag{3}$$
so, for example, $q_{50}=q_0+50$ whence $p_{50}=\left(\frac{1}{p_0}+50\right)^{-1}$.

