Let `X_(n)` denote the mean of first n natural numbers, then the mean of `X_(1), X_(2), ………….., X_(100)` is

Options

A. 25
B. 50
C. 25.5
D. 25.75

Answer

To find the mean of \( X_1, X_2, \ldots, X_{100} \), where \( X_n \) denotes the mean of the first \( n \) natural numbers, we can follow these steps:

### Step 1: Calculate \( X_n \)
The mean of the first \( n \) natural numbers is given by the formula:
\[ X_n = \frac{1 + 2 + 3 + \ldots + n}{n} \]
The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Thus, we have:
\[ X_n = \frac{\frac{n(n + 1)}{2}}{n} = \frac{n + 1}{2} \]

### Step 2: Calculate \( X_1, X_2, \ldots, X_{100} \)
Now we can calculate \( X_1, X_2, \ldots, X_{100} \):
- \( X_1 = \frac{1 + 1}{2} = 1 \)
- \( X_2 = \frac{2 + 1}{2} = 1.5 \)
- \( X_3 = \frac{3 + 1}{2} = 2 \)
- \( X_4 = \frac{4 + 1}{2} = 2.5 \)
- ...
- \( X_{100} = \frac{100 + 1}{2} = 50.5 \)

In general, we can express:
\[ X_n = \frac{n + 1}{2} \]
for \( n = 1, 2, \ldots, 100 \).

### Step 3: Calculate the Mean of \( X_1, X_2, \ldots, X_{100} \)
Now we need to find the mean of these values:
\[ \text{Mean} = \frac{X_1 + X_2 + \ldots + X_{100}}{100} \]
Substituting the expression for \( X_n \):
\[ \text{Mean} = \frac{\sum_{n=1}^{100} \frac{n + 1}{2}}{100} \]
This can be simplified as:
\[ \text{Mean} = \frac{1}{2} \cdot \frac{\sum_{n=1}^{100} (n + 1)}{100} \]
\[ = \frac{1}{2} \cdot \frac{\sum_{n=1}^{100} n + \sum_{n=1}^{100} 1}{100} \]
\[ = \frac{1}{2} \cdot \frac{\frac{100(100 + 1)}{2} + 100}{100} \]
\[ = \frac{1}{2} \cdot \frac{5050 + 100}{100} \]
\[ = \frac{1}{2} \cdot \frac{5150}{100} \]
\[ = \frac{1}{2} \cdot 51.5 = 25.75 \]

### Final Answer
The mean of \( X_1, X_2, \ldots, X_{100} \) is:
\[ \boxed{25.75} \]