In a data the number I is repeated I times for i=1, 2, …,n. Then the mean of the data is

Options

A. `(2n+1)/(6)`
B. `(2n+1)/(4)`
C. `(2n+1)/(3)`
D. `(2n+1)/(2)`

Answer

To find the mean of the data where the number \( i \) is repeated \( i \) times for \( i = 1, 2, \ldots, n \), we can follow these steps:

### Step 1: Understand the Data Structure
The data consists of the number \( i \) repeated \( i \) times. This means:
- The number 1 appears 1 time,
- The number 2 appears 2 times,
- The number 3 appears 3 times,
- ...
- The number \( n \) appears \( n \) times.

### Step 2: Calculate the Total Number of Observations
The total number of observations can be calculated as:
\[ \text{Total observations} = 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]

### Step 3: Calculate the Sum of the Data
The sum of the data can be calculated as:
\[ \text{Sum} = 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3 + \ldots + n \cdot n = 1^2 + 2^2 + 3^2 + \ldots + n^2 \]
Using the formula for the sum of squares:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 4: Calculate the Mean
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of the data}}{\text{Total observations}} = \frac{\frac{n(n + 1)(2n + 1)}{6}}{\frac{n(n + 1)}{2}} \]
Simplifying this expression:
\[ \text{Mean} = \frac{n(n + 1)(2n + 1)}{6} \cdot \frac{2}{n(n + 1)} = \frac{2(2n + 1)}{6} = \frac{2n + 1}{3} \]

### Conclusion
Thus, the mean of the data is:
\[ \text{Mean} = \frac{2n + 1}{3} \]

### Final Answer
The mean of the data is \( \frac{2n + 1}{3} \).

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