Nested for loop in Big Oh Complexity

Question

for(int i = 0; i < n; i++) {
    for(int j = 0; j < i; j++){
        //do swap stuff, constant time
    }
}

I read that single for loop is O(N) a

Answer 1

As you mentioned

Each takes unit of 1 - O(1)

So each iteration of the inner loop takes 1, 2, 3, ... , n unit time.

total_time = 1 +   2   +   3   + ... + (n-2) + (n-1) + n

Reversing

total_time = n + (n-1) + (n-2) + ... +   3   +   2   + 1

Adding

2 * total_time = (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1) + (n+1)

There are total n terms

2 * total_time = (n+1) * n

=> total_time = (n+1) * n / 2

=> total_time = n^2 / 2 + n / 2

Lower terms and constant coefficients are neglected for big O complexity.

As a result

O(n^2)