#include <iostream>
using namespace std;

int sum(int a) {
    return ((1 + a)/2)*a;
}

int main () {
    int a;

    cin >> a;

    if (a > 0) cout << sum(a);
    else if (a < 0) cout << sum(abs(a)) * -1 + 1;
    else cout << 0;

    return 0;
}

if N = 0 then you need add one
so the correct solution is:

if (a > 0) cout << sum(a);
    else if (a < 0) cout << sum(abs(a)) * -1 + 1;
    else cout << 1;