C

Step 1: Let the height of the tower be H and the initial velocity be u. For the upward motion, the time taken to reach the highest point ( t_{up} ) is calculated using the formula: t_{up} = \frac{u}{g}, where g is the acceleration due to gravity.
Step 2: The total time of flight (t_total) comprises the time taken to ascend and the time taken to descend. According to the problem, t_total = n × t_{up}.
Step 3: The time to fall back to the ground can be determined from the equation of motion: H = \frac{1}{2}g(t_{down})^2. Given that t_{down} = t_total - t_{up}, we can set up the equation:
H = \frac{1}{2}g(n\frac{u}{g} - \frac{u}{g})^2.
By simplifying and solving the quadratic relationship for n, we find the equation matches with Option C: \(2gH = n^{2}u^{2}\).
Therefore, C.