The standard deviation of first 50 even natural number is ` lamda ` then value of ` [lamda] ` is ( where [.] is gretest interger )…

Explanation

To find the standard deviation of the first 50 even natural numbers, we can follow these steps:

### Step 1: Identify the first 50 even natural numbers
The first 50 even natural numbers are:
\[ 2, 4, 6, \ldots, 100 \]

### Step 2: Calculate the mean of these numbers
The mean \( \mu \) of the first \( n \) even natural numbers can be calculated using the formula:
\[ \mu = \frac{\text{Sum of the first } n \text{ even natural numbers}}{n} \]
The sum of the first \( n \) even natural numbers is given by:
\[ \text{Sum} = n(n + 1) \]
For \( n = 50 \):
\[ \text{Sum} = 50 \times 51 = 2550 \]
Thus, the mean is:
\[ \mu = \frac{2550}{50} = 51 \]

### Step 3: Calculate the sum of squares of these numbers
We need to calculate the sum of the squares of the first 50 even natural numbers:
\[ \text{Sum of squares} = 2^2 + 4^2 + 6^2 + \ldots + 100^2 \]
This can be expressed as:
\[ \text{Sum of squares} = 4(1^2 + 2^2 + 3^2 + \ldots + 50^2) \]
Using the formula for the sum of squares:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]
For \( n = 50 \):
\[ \sum_{k=1}^{50} k^2 = \frac{50 \times 51 \times 101}{6} = 42925 \]
Thus:
\[ \text{Sum of squares} = 4 \times 42925 = 171700 \]

### Step 4: Calculate the variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\text{Sum of squares}}{n} - \mu^2 \]
Substituting the values we have:
\[ \sigma^2 = \frac{171700}{50} - 51^2 \]
Calculating:
\[ \sigma^2 = 3434 - 2601 = 833 \]

### Step 5: Calculate the standard deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{833} \approx 28.844 \]

### Step 6: Find the greatest integer less than or equal to \( \sigma \)
The greatest integer less than or equal to \( 28.844 \) is:
\[ \lfloor \sigma \rfloor = 28 \]

### Final Answer
Thus, the value of \( [\lambda] \) is:
\[ \boxed{28} \]