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Step-by-Step Solution
Step 1: Define the times involved
Let the time for the particle to go from the initial throw up to its highest point be $t_{1}$. Let the total time taken to hit the ground (from the instant it was thrown) be $T$. According to the problem,
$$
T = n \, t_{1}.
$$
Step 2: Find $t_{1}$, the time to reach the highest point
When a particle is thrown vertically upwards with initial speed $u$ (accelerated downward by $g$), the velocity at the highest point is zero. Using
$$
v = u - g\,t_{1}, \quad \text{with } v=0,\,
$$
we get
$$
0 = u - g\,t_{1} \quad \Longrightarrow \quad t_{1} = \frac{u}{g}.
$$
Step 3: Express the total time $T$
From the given condition $T = n\,t_{1}$,
$$
T = n \times \frac{u}{g} = \frac{n\,u}{g}.
$$
Step 4: Separate the motion into two parts
Upward journey: from the tower top to the highest point (time = $t_{1} = \tfrac{u}{g}$).
Downward journey: from the highest point down to the ground (time = $t_{2}$).
The total time is $T = t_{1} + t_{2}$. Hence,
$$
t_{2} = T - t_{1} = \frac{n\,u}{g} - \frac{u}{g} = \frac{(n - 1)u}{g}.
$$
Step 5: Calculate $t_{2}$ in an alternate way
At the highest point, the particle’s velocity is zero. The distance from the highest point to the ground is the initial tower height $H$ plus the extra height gained above the tower. The extra height gained (from tower top to highest point) is
$$
\frac{u^2}{2g}.
$$
Thus, the total vertical distance from the highest point down to the ground is
$$
H + \frac{u^2}{2g}.
$$
Since it starts from rest at the highest point (going downward under gravity), we use
$$
H + \frac{u^2}{2g} = \frac{1}{2} g\,t_{2}^2.
$$
Step 6: Equate the two expressions for $t_{2}$
From the distance equation,
$$
t_{2} = \sqrt{\frac{2\Bigl(H + \tfrac{u^2}{2g}\Bigr)}{g}}
= \sqrt{\frac{2H + \frac{u^2}{g}}{g}}
= \frac{1}{g}\sqrt{2\,H\,g + u^2}.
$$
From the time relation found earlier,
$$
t_{2} = \frac{(n - 1)\,u}{g}.
$$
Equating these,
$$
\frac{(n - 1)\,u}{g} = \frac{1}{g}\sqrt{2\,H\,g + u^2}.
$$
Step 7: Solve for the required relationship
Multiply both sides by $g$:
$$
(n - 1)\,u = \sqrt{2\,H\,g + u^2}.
$$
Square both sides:
$$
(n - 1)^2\,u^2 = 2\,H\,g + u^2.
$$
Rearrange to isolate $2\,H\,g$:
$$
2\,H\,g = (n - 1)^2\,u^2 - u^2 = u^2\bigl((n - 1)^2 - 1\bigr) = u^2\bigl(n^2 - 2n\bigr).
$$
Factor out $n$:
$$
2\,g\,H = u^2\,n\,(n - 2).
$$
Therefore,
$$
\boxed{2\,g\,H = n\,u^2\,(n - 2).}
$$
Final Answer
The correct relation between $H$, $u$, and $n$ is
$$
2gH = n \, u^2 \,(n - 2).
$$
