1. Let the initial speed with which each ball is thrown be $u$.

2. Time taken by a ball to reach the highest point is $t=\frac{u}{g}.$

3. According to the question, when the first ball reaches its highest point, the juggler throws the next ball. So the time interval between successive throws is $\Delta t=\frac{u}{g}.$

4. The juggler throws $n$ balls per second. Hence the time between two throws is also $\Delta t=\frac{1}{n}.$

5. Equating the two expressions for the time interval, $\frac{u}{g}=\frac{1}{n}$ which gives $u=\frac{g}{n}.$

6. Maximum height reached by a vertically thrown ball is $H=\frac{u^2}{2g}.$ Substitute $u=\frac{g}{n}$: H=\frac{(g/n)^2}{2g}=rac{g^2}{2gn^2}=rac{g}{2n^2}.

7. Therefore, the maximum height is $\boxed{\frac{g}{2n^2}}.$

8. Checking options:

• A: $\frac{g}{2n}$ — incorrect
• B: $\frac{g}{n}$ — incorrect
• C: $2gn$ — incorrect
• D: $\frac{g}{2n^2}$ — correct

Hence the correct option is D.