while lo + 1 <= r:
   lo += 1
   sum += a[lo]

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!<{margin-right: 20px; auto;display:block;border: 1px solid gray;}blog/square-root-trick?image001.png!
 
 

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The query takes less than <tex>k + n/k + k = 2k + n/k</tex> time. 2k + n/k is minimized when k ~ sqrt(n). For k = sqrt(n) the query takes O(sqrt(n)) time.
This trick also works for other associative operations, like: min, gcd, product etc.