Finding an Explicit Formula for Probability Given the Recursive Formula I am attempting to solve a problem asking for the formula to find a probability, $p(n)$, $n$ being 3, 4, 5,... I was able to determine that $p(3) = 1/4$  and the $p(n)$ given not $p(n-1)= \frac{n-2}{2(n-1)}$. From that, I can write the recursive formula: 
$$p(n) = p(n-1) + \frac{(1-p(n-1))(n-2)}{2(n-1)}$$
How can I turn this into an explicit formula for $p(n)$? Thanks. 

I will post a solution by @robjohn
Let $t(n) = 1 - p(n)$, then we rewrite the equation as
$$ t(n) = \frac{n}{2(n-1)} t(n-1) = \frac{n}{2(n-1)}\cdot\frac{n-1}{2(n-2)}\cdot t(n-2)$$
you proceed to a generic step $k$
$$t(n) = \frac{n}{2(n-1)}\cdot\frac{n-1}{2(n-2)}\cdot\ldots\cdot\frac{n-k+1}{2(n-k)}\cdot t(n-k) = \frac{n}{2^k (n-k)}\cdot t(n-k)$$
Now, substitute $k = n-3$:
$$t(n) = \frac{n}{2^{n-3} \cdot 3}\cdot t(3) = \frac{n}{2^{n-3} \cdot 3} \cdot \frac{3}{4} = \frac{n}{2^{n-1}}$$
Therefore, $$p(n) = \frac{2^{n-1}-n}{2^{n-1}}.$$
