To solve this problem, we need to find a pattern in the series and then find the value of x that satisfies the equation.

Let's first rewrite the series:

1/1.2 + 1/2.3 + 1/3.4 + ... + 1/x(x+1) = 2002/2003

Notice that each term in the series can be written as:

1/n(n+1) = 1/n - 1/(n+1)

Therefore, the series can be rewritten as:

(1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/x - 1/(x+1))

This simplifies to:

1 - 1/(x+1) = 2002/2003

Now, we can solve for x:

1 - 1/(x+1) = 2002/2003
1 - 2002/2003 = 1/(x+1)
1/2003 = 1/(x+1)
x + 1 = 2003
x = 2002

Therefore, the value of x that satisfies the equation is 2002.