Why Nested Loops are O(n^2)

When you have two nested loops both running from $1$ to $n$, the total operations is $n + n + ... + n$ ($n$ times), which equals $n^2$.

But what if the inner loop runs from $i$ to $n$? Then you compute $n + (n-1) + (n-2) + ... + 1$. Using the formula, this equals $\frac{n(n+1)}{2}$, which is $O(n^2)$.

The coefficient changes ($\frac{1}{2}$ instead of $1$), but the dominant term is still $n^2$. This is why checking all pairs in an array takes quadratic time.