Does analytical solution for $a_i$ exist?

Question

For the system of $n$ equations ($i = 1, 2, ..., n)$ of the following kind: $$P_i = \sum_{j=0}^{i-1} a_j + (7a_i+a_{i+1})/8$$ And assuming $a_1 = 0$ and $a_{n+1} = 0$, is there an analytical solution for $a_i$? If so, what is it?

Answer 1

If the indices are right, then this is a lower triangular linear system on the unknowns $a_2, \dots, a_n$, whose diagonal entries are all equal to $1/8$. Therefore, it has a unique solution, which can be found by forward substitution. It is easy to right a formula for the solution.