Problem 1: Recursive description of the value function

Find a recurrence relation satisfied by the value function. This relation should express the value function for column k, namely the values C(k, j) for j=1..m, in terms of the values of the value function for the previous column, $C(k-1, \cdot)$,together with the weight values $\text{cost}((k-1, \cdot),(k, j))$of the arcs from nodes in column k-1 to each given node in column k. Here, k ranges from 1 to n+1.