Consider the sequence $a_0, a_1, ..., a_{2n}$, where $a_0=0$, $a_{2n}=1$, and $a_i=wa_{i-1}+(1-w)a_{i+1}$ for all indexes $i\in [1, 2n-1]$. In other words, the first term of the sequence is 0, the last term is 1, and term in between is the weighted average of its adjacent terms.
In terms of $w$, what is $a_n$ (the middle term)? And a more general question: what is the explicit formula for each $a_i$?
Look at the cases for $n=1, 2, 3$, it seems that $a_n=\frac{w^n}{w^n+(1-w)^n}$, but unfortunately I don't know how to prove it for general $n$. Some other things I have tried to do so far is to look at it recursively and directly compute, but the terms get very ugly quickly. I would appreciate any help on this problem.