If `(1)/(x+1) + (2)/(y+2) + (1009)/(z+ 1009)=1` find `(x)/(x+1) + (y)/(y+2) + (z)/(z + 1009)`= ?

Answer

To solve the problem, we start with the given equation:

\[ \frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1 \]

We need to find the value of:

\[ \frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009} \]

Let's denote:

\[ t = \frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009} \]

### Step 1: Rewrite the terms
We can rewrite each term in \(t\):

\[ \frac{x}{x+1} = 1 - \frac{1}{x+1} \]
\[ \frac{y}{y+2} = 1 - \frac{2}{y+2} \]
\[ \frac{z}{z+1009} = 1 - \frac{1009}{z+1009} \]

### Step 2: Substitute into \(t\)
Now substituting these back into \(t\):

\[ t = \left(1 - \frac{1}{x+1}\right) + \left(1 - \frac{2}{y+2}\right) + \left(1 - \frac{1009}{z+1009}\right) \]

This simplifies to:

\[ t = 3 - \left(\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009}\right) \]

### Step 3: Use the given equation
From the original equation, we know:

\[ \frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1 \]

Substituting this into the equation for \(t\):

\[ t = 3 - 1 \]

### Step 4: Final calculation
Thus, we find:

\[ t = 2 \]

### Conclusion
The value of

\[ \frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009} = 2 \]