I agree with the already posted answers. However, I would like to present an alternative way of obtaining an asymptotic lower bound without having to obtain the closed form.
Consider only cases of
n>100. For over
c is both greater than
n/3 and less than
2n/3. Each of those iterations of the outer loop does at least
n/3 iterations of the middle loop, each of which does at least
n/3 iterations of the innermost loop. Thus the total number of innermost loop iterations is at least
(n/4) * (n/3) * (n/3).
The number of innermost loop iterations is bounded above by
n^3, because each of the three nested loops does no more than
A function that is bounded below by
(n^3)/36 and above by