Real root of the equation
`(x-1)^(2013) +(x-2)^ (2013) +(x-3)^ (2013)+……….+(x-2013)^(2013) =0` is a four digit number. Then the sum of the digits is :

Answer

To find the real root of the equation

\[ (x-1)^{2013} + (x-2)^{2013} + (x-3)^{2013} + \ldots + (x-2013)^{2013} = 0, \]

we can follow these steps:

### Step 1: Identify the Middle Value
The terms in the equation range from \(1\) to \(2013\). The middle term can be calculated as:

\[ \text{Middle term} = \frac{\text{First term} + \text{Last term}}{2} = \frac{1 + 2013}{2} = \frac{2014}{2} = 1007. \]

### Step 2: Substitute the Middle Value
Now, we will substitute \(x = 1007\) into the equation:

1. The first term becomes:
\[ (1007 - 1)^{2013} = (1006)^{2013}. \]

2. The last term becomes:
\[ (1007 - 2013)^{2013} = (-1006)^{2013}. \]

### Step 3: Analyze the Terms
Notice that \( (-1006)^{2013} \) is the negative of \( (1006)^{2013} \). Therefore, the first and last terms cancel each other out:

\[ (1006)^{2013} + (-1006)^{2013} = 0. \]

### Step 4: Check Other Terms
Next, we check the second term and the second last term:

1. The second term becomes:
\[ (1007 - 2)^{2013} = (1005)^{2013}. \]

2. The second last term becomes:
\[ (1007 - 2012)^{2013} = (-1005)^{2013}. \]

Again, these two terms cancel each other out:

\[ (1005)^{2013} + (-1005)^{2013} = 0. \]

### Step 5: Continue This Pattern
Continuing this pattern, we see that all terms will cancel out in pairs, leaving us with:

\[ (1007 - 1007)^{2013} = 0^{2013} = 0. \]

Thus, \(x = 1007\) is indeed a root of the equation.

### Step 6: Find the Sum of the Digits
Now, we need to find the sum of the digits of \(1007\):

\[ 1 + 0 + 0 + 7 = 8. \]

### Final Answer
The sum of the digits is:

\[ \boxed{8}. \]

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