Find an asymptotic bound for $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+...+T(\frac{n}{2^k})$
Assuming an appropriate base case, you can prove that $T(n) \leq \frac{3}{2} n^2$. (If this doesn't work for the base case, just increase $3/2$ to a large enough constant.)

Indeed, assuming this holds for $m &lt; n$, we have
$$
\begin{align*}
T(n) &amp;= n^2 + T(\tfrac{n}{2}) + T(\tfrac{n}{4}) + \cdots + T(\tfrac{n}{2^k}) \\ &amp;\leq
n^2 + \frac{3}{2} \frac{n^2}{4} + \frac{3}{2} \frac{n^2}{4^2} + \cdots + \frac{3}{2} \frac{n^2}{4^k} \\ &amp;=
n^2 \left(1 + \frac{3}{2} \left(\frac{1}{4} + \frac{1}{4^2} + \cdots + \cdots + \frac{1}{4^k}\right)\right) \\ &amp;&lt;
n^2 \left(1 + \frac{3}{2} \cdot \frac{1}{3}\right) \\ &amp;= \frac{3}{2} n^2.
\end{align*}
$$