cses Range Updates and Sums
The idea is to use a lazy segment tree to get the sum of ranges in log N time complexity.
int A[N];
long long t[N << 2];
The standard segment tree idea is that the sum of a vertex’s segment is equal to the sum of vertex’s children’s segments. This is made clear in a segment tree’s construction.
void build(int i, int l, int r) {
if (l == r) {
t[i] = A[l];
}
else {
int m = l + (r  l) / 2;
build(i<<1, l, m);
build(i<<11, m + 1, r);
t[i] = t[i<<1] + t[i<<11];
}
}
However, an eager as opposed to lazy range update on a segment tree might affect many ranges, degrading the log N time complexity of operations. For example, adding 1 to every element in the array would require the entire tree to be calculated again.
You can think of a lazy update to a range as completed for that range, but pending for its subrange. Prior to any descent into those subranges, you should propagate the changes.
For example, say I add 1 to all values in the array: I can increase the segment tree’s root sum and remember that 1 was added to all values with a lazy tag. Any reads to that exact range will just return the range’s sum. But now what if I want the sum of a range within the one I just lazily updated? Look below.
Ignore the set operation for now. Notive the new lazy add member added to each vertex.
struct node {
ll sum;
ll lz_add;
node() {}
} t[N << 2];
void add(int i, int l, int r, int a, int b, ll x) {
if (a > b) {
return;
}
else if (l == a && r == b) {
t[i].lz_add += x;
t[i].sum += x * (r  l + 1);
}
else {
// 1. what should be done right here?
int m = l + (r  l) / 2;
add(i<<1, l, m, a, min(b, m), x);
add(i<<11, m + 1, r, max(a, m + 1), b, x);
// 2. and what should be done here?
}
}

Prior to any descent into those subranges, you have to propagate the changes you made and clear the lazy tag.

Since you are going to update a lower range lazily as well, the segment tree invariant needs to be upholded.
item 2 is quite simple:
t[i] = t[i<<1] + t[i<<11];
I will leave item 1 to you. Look at my submission or the usaco guide if you are confused. USACO guide