The query is interesting. The elements of the first and last slice (partially contained in the queried range) have to be traversed one by one, but for slices completely contained in our range we can use the values in $S$ directly and get a performance boost.
Here is an update example:

p. !<{margin-right: 20px; auto;display:block;}blog/square-root-trick?image01.png!
p. In $update(2, 5)$ we have to change $A[2]$ to 5 which results in changing the value of $S[1]$ to keep $S$ up to date.

!<{margin-right: 20px; auto;display:block;}blog/square-root-trick?image01.png!
In $update(2, 5)$ we have to change $A[2]$ to 5 which results in changing the value of $S[1]$ to keep $S$ up to date.

p. !<{margin-right: 20px; auto;display:block;}blog/square-root-trick?image00.png!

!<{margin-right: 20px; auto;display:block;}blog/square-root-trick?image00.png!

In query(2, ) we get A[2] + A[3] + (A[4] + ...

In $query(2, )$ we get $A[2] + A[3] + (A[4] + ...$

p. The code looks like this: