Given:
$T(n) = 7T(n/2) + n^3$
Compare with:
$T(n) = aT(n/b) + f(n)$
Here:
$a = 7$
$b = 2$
$f(n) = n^3$
Now calculate:
$n^{\log_b a} = n^{\log_2 7}$
Since:
$\log_2 7 \approx 2.807$
So:
$n^{\log_2 7} = n^{2.807}$
Now compare $f(n)$ with $n^{\log_b a}$:
$f(n) = n^3$
$n^3$ grows polynomially faster than $n^{2.807}$.
So this is Master Theorem Case 3.
Therefore:
$T(n) = \Theta(f(n)) = \Theta(n^3)$
Final answer: $\boxed{\Theta(n^3)}$
