We want to find the value of $X$ in the given situation.
We are given that $a_1 = X$ and $a_n = \frac{2S_n^2}{2S_n - 1}$ for all $n \ge 2$.
To find $a_{100}$, we need to determine the value of $S_{100}$.
We can start by finding the value of $S_2$ using the given information:
$a_2 = \frac{2S_2^2}{2S_2 - 1}$
Substituting $n = 2$ into the equation, we have:
$S_2 = a_1 + a_2 = X + \frac{2S_2^2}{2S_2 - 1}$
Multiplying both sides of the equation by $2S_2 - 1$ to eliminate the fraction, we get:
$(2S_2 - 1)S_2 = X(2S_2 - 1) + 2S_2^2$
Expanding and simplifying, we have:
$2S_2^2 - S_2 = 2XS_2 - X + 2S_2^2$
Combining like terms, we get:
$X = S_2$
Substituting this value into the equation for $a_n$, we have:
$a_n = \frac{2S_n^2}{2S_n - 1} = \frac{2X^2}{2X - 1}$
Now, we know that $a_{100} = \frac{2S_{100}^2}{2S_{100} - 1}$. Given that the value of $a_{100}$ is $-\frac{2}{39203}$, we can write:
$-\frac{2}{39203} = \frac{2S_{100}^2}{2S_{100} - 1}$
Multiplying both sides of the equation by $(2S_{100} - 1)$ to eliminate the fraction, we get:
$-2 = 2S_{100}^2$
Dividing both sides of the equation by 2, we find:
$-1 = S_{100}^2$
Taking the square root of both sides of the equation, we have:
$S_{100} = \pm 1$
Since $S_{100}$ is a sum of terms in the sequence, it must be positive. Therefore, we can write:
$S_{100} = 1$
Substituting this value into the equation $X = S_2$, we find:
$X = 1$
The value of X is 1.
The answer is: 1
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.

If $a_1 = X and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}.$
If we know the answer to the above question is -\frac{2}{39203}, what is the value of unknown variable X?

### Answer:
