class NumArray {
    int[] prefix;

    // arr was 1-indexed, but nums is 0-indexed.
    public NumArray(int[] nums) {
        int n = nums.length;

        // `prefix` array is 1-indexed because it greatly simplifies computations.
        // If `prefix` array would be 0-indexed, we'd have to always consider the case
        // when i = 0 or left = 0 separately, so that we don't go out-of-bounds.
        prefix = new int[n + 1]; // prefix[0 .. n]
        
        //            0  1  2   3  4   5
        // nums =   [-2, 0, 3, -5, 2, -1]

        //            0   1   2   3   4   5   6
        // prefix = [ 0, -2, -2,  1, -4, -2, -3]

        // prefix[0] = 0 because it is semantically equal to an empty prefix (which has zero sum).
        prefix[0] = 0;
        for (int i = 1; i <= n; i++) {
            // prefix[i] = 
            //             sum(nums[1 .. i - 1]) +
            //             nums[i]
            prefix[i] = prefix[i - 1] + nums[i - 1];

            // nums[i - 1] (not nums[i]) because `nums` array is 0-indexed.
        }
    }

    public int sumRange(int left, int right) {
        // Increment by 1 because `prefix` array is 1-indexed.
        left++; right++;

        // nums[left] + ... nums[right]
        // = 
        // sum(nums[1 .. right]) -
        // sum(nums[1 .. left - 1]) = 
        // sum(nums[left .. right])
        return prefix[right] - prefix[left - 1]; 
    }
}

/**
 * Your NumArray object will be instantiated and called as such:
 * NumArray obj = new NumArray(nums);
 * int param_1 = obj.sumRange(left,right);
 */