Isolating Variables in Loops

Sometimes you need to solve for loop bounds algebraically. For example, you want a loop that runs while $k^2 \leq n$.

Solve for $k$: $k \leq \sqrt{n}$. So the loop runs from $k = 1$ to $k = \lfloor \sqrt{n} \rfloor$. This gives $O(\sqrt{n})$ iterations.

Another example: a loop doubles $k$ each iteration until $k \geq n$. Solve $2^i = n$ to get $i = \log_2 n$, meaning $O(\log n)$ iterations.