If `{x} and [x]` represent fractional and integral part of x, then `[x]+ sum_(r=1)^(1090)({x+r})/(1090)=`

Options

A. x
B. 1090x
C. `(x)/(1090)`
D. 1090

Answer

To solve the problem, we need to find the value of the expression:

\[ [x] + \frac{\sum_{r=1}^{1090} \{x + r\}}{1090} \]

where \([x]\) is the integral part of \(x\) and \(\{x\}\) is the fractional part of \(x\).

### Step-by-Step Solution:

1. **Understanding the Components**:
- We know that \(x\) can be expressed as:
\[ x = [x] + \{x\} \]
- Here, \([x]\) is the integral part and \(\{x\}\) is the fractional part of \(x\).

2. **Evaluating the Summation**:
- We need to evaluate the summation:
\[ \sum_{r=1}^{1090} \{x + r\} \]
- The fractional part \(\{x + r\}\) can be expressed as:
\[ \{x + r\} = \{[x] + \{x\} + r\} = \{r + \{x\}\} \]
- Since \([x]\) is an integer, it does not contribute to the fractional part.

3. **Breaking Down the Summation**:
- The fractional part \(\{r + \{x\}\}\) can be simplified. For each \(r\):
- If \(r + \{x\} < 1\), then \(\{r + \{x\}\} = r + \{x\}\).
- If \(r + \{x\} \geq 1\), then \(\{r + \{x\}\} = (r + \{x\}) - 1\).
- This means that for \(r = 1\) to \(1090\), we can consider how many times \(\{x\}\) contributes to the fractional part.

4. **Calculating the Total**:
- The sum can be split into two parts based on the value of \(\{x\}\):
- For \(r\) such that \(r + \{x\} < 1\) (which happens for small values of \(r\)), the contribution is simply \(\{x\}\).
- For larger \(r\), the contribution will be \((r + \{x\}) - 1\).
- However, since we are summing over a large range, we can approximate the total contribution.

5. **Final Calculation**:
- The total number of terms is \(1090\), and the average contribution from the fractional parts will yield:
\[ \sum_{r=1}^{1090} \{x + r\} \approx 1090 \cdot \{x\} \]
- Therefore, the average becomes:
\[ \frac{\sum_{r=1}^{1090} \{x + r\}}{1090} \approx \{x\} \]

6. **Putting It All Together**:
- Now substituting back into the original expression:
\[ [x] + \frac{\sum_{r=1}^{1090} \{x + r\}}{1090} = [x] + \{x\} = x \]

### Conclusion:
Thus, the value of the expression is:

\[ \boxed{x} \]