Range updates through differences

Range add, point query. Adding v to [l, r] is upd(l, v) and upd(r+1, -v) on a plain Fenwick of the difference array; the value at position i is prefix(i). O(log n) per operation, ten lines total.

Range add, range sum. The prefix sum of a range-updated array needs two BITs holding the coefficients of an expanded formula. After adding v to [l, r]:

sum(1..i) = i * B1(i) - B2(i)
B1: upd(l, v), upd(r+1, -v)
B2: upd(l, v*(l-1)), upd(r+1, -v*r)

Derivation: write the array as a sum of ramps (the difference array integrated once). If a problem mixes range-add with range-sum but has no min/max or assign, this is lighter than a lazy segment tree — fewer invariants, smaller constants, fewer bugs.

When NOT to bother: any operation pair involving min/max or assignment breaks differences — go straight to lazy propagation (Module 2).