To solve the problem of finding the missing element in an array `A`

consisting of `N`

different integers from the range `[1..(N + 1)]`

, you can utilize the mathematical approach of calculating the expected sum versus the actual sum of the array elements.

Here’s the step-by-step explanation of the approach:

The sum of the first `N+1`

integers (which are `[1, 2, ..., N+1]`

) can be calculated using the formula:

This formula gives the sum of all integers from `1`

to `N+1`

.

Compute the sum of all elements in array `A`

.

The missing element will be the difference between the expected sum and the actual sum of array `A`

.

Given the constraints (N can be up to 100,000), this approach is efficient with a time complexity of O(N) because it involves a single pass through the array to compute the sum.

Here is the implementation of the solution in JavaScript:

```
function solution(A) {
const N = A.length;
const expectedSum = (N + 1) * (N + 2) / 2;
let actualSum = 0;
for (let i = 0; i < N; i++) {
actualSum += A[i];
}
const missingElement = expectedSum - actualSum;
return missingElement;
}
// Example usage:
const A = [2, 3, 1, 5];
console.log(solution(A)); // Output: 4
```

### Explanation of the Example:

For the array `A = [2, 3, 1, 5]`

, where `N = 4`

(since `A.length = 4`

), the expected sum of `[1, 2, 3, 4, 5]`

is `15`

.

The actual sum of elements in `A`

is `2 + 3 + 1 + 5 = 11`

.

Therefore, the missing element is `15 - 11 = 4`

.

This approach ensures that you find the missing element in linear time, making it optimal for large inputs as specified in the problem constraints.