This problem can be solved in a few ways, generally revolving around data structures that support range update with lazy propagation and range query. We'll outline one such solution that uses two segment trees.

We'll take one segment tree that maintains an array ~A~ of elements and supports range increment and the range sum query ~f(l, r) = \sum_{i=l}^r A[i]~. Another one will support range increment and the range sum query ~f(l, r) = \sum_{i=l}^r i \cdot A[i]~. The original range sum query will be answered by summing the range sum query over both segment trees. To perform a range increment, perform a range increment on the first segment tree over ~[l, r]~ with value ~(1-l) \cdot a~, and perform a range increment on the second segment tree over ~[l, r]~ with value ~a~.