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This approach involves iterating through each number from 1 to n and checking if it satisfies the given condition of a pivot integer. For each candidate pivot integer x, calculate the sum of numbers from 1 to x using the formula for the sum of an arithmetic series. Similarly, calculate the sum from x to n. Compare these sums to determine if x is a pivot integer.
Time Complexity: O(n).
Space Complexity: O(1).
1function findPivot(n) {
2 for (let x = 1; x <= n; ++x) {
3 let sum1 = x * (x + 1) / 2;
4 let sum2 = (n * (n + 1) / 2) - ((x - 1) * x / 2);
5 if (sum1 === sum2) return x;
6 }
7 return -1;
8}
9
10let n = 8;
11console.log(findPivot(n));The JavaScript function performs similar steps as the above languages. It checks pivot x by verifying equality of the expected sums.
Instead of iterating through each possible x, we can use mathematical formulation. As the sum from 1 to n is fixed, let's denote the total sum as S. Then, for pivot x, the condition becomes:
(x * (x + 1)) / 2 = (S - ((x - 1) * x) / 2)
From this, derive a formula to find x directly if it exists. Solving algebraically can help us identify the pivot integer without iteration.
Time Complexity: O(n).
Space Complexity: O(1).
1
In Python, this solution simplifies finding the pivot by checking an optimized mathematical equality for each x.