Numeric center in Java
About the code itself(not the algorithm used):


Class name usually starts with a capital letter in Java.
Variables' names: it is better to use more meaningful names. For example, a name firstSum is much clearer then fsum. 
Variable declaration and initialization: it is a good practice to make a scope of each variable as narrow as possible. It means that fsum and ssum should be declared inside the loop's body.
If the number of iterations of a loop is known, it makes more sense to use for loop then do ... while. Conversely, when the number of iterations is not known, using while is better than using for and break. 
Surrounding operators by whitespaces makes the code more readable:

for (int j = index + 1; j &lt; index * 2; j++) 


looks better than

for (int j=index+1;j&lt;index*2;j++){

You should use proper and consistent indentation(closing bracket indented by 4 spaces looks strange).
I would declare a magic number(204) as a final static field of this class, not as a local variable(and a name n is not good for it because it is not meaningful).


So an improved version of your code can look like this:

public class NumericalCenter {

    private final static int MAX_CANDIDATE = 204;

    public static void main(String args[]) {
        for (int candidate = 1; candidate &lt;= MAX_CANDIDATE; candidate++) {
            int firstSum = 0;
            for (int i = 1; i &lt; candidate; i++)
                firstSum += i;
            int secondSum = 0;
            int i = candidate + 1;
            while (secondSum &lt; firstSum) {
                secondSum += i;
                i++;
            }
            if (firstSum == secondSum)
                System.out.println(candidate + " is a numerical center");
        }
    }
}


Now about the algorithm itself: let's assume that a candidate number is fixed(let's call it x). Then we get an equation:
$${n(n+1) \over 2} - {x(x+1) \over 2} = {x(x-1) \over 2}$$ 

It is a quadratic equation(with respect to n), so it can be solved in O(1)(which gives O(MAX_CANDIDATE) time complexity in total). However, it is not necessary for MAX_CANDIDATE = 204.