What is the value of n for which the numbers 1, 2, 3, …., n have variance 2?

Options

A. 4
B. 5
C. 6
D. 8

Explanation

To find the value of \( n \) for which the numbers \( 1, 2, 3, \ldots, n \) have a variance of 2, we will follow these steps:

### Step 1: Understand the formula for variance
The variance \( \sigma^2 \) of a set of numbers is given by the formula:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2 \]
where \( \mu \) is the mean of the numbers and \( n \) is the total number of observations.

### Step 2: Calculate the mean \( \mu \)
The mean \( \mu \) of the first \( n \) natural numbers is calculated as:
\[ \mu = \frac{1 + 2 + 3 + \ldots + n}{n} = \frac{\frac{n(n + 1)}{2}}{n} = \frac{n + 1}{2} \]

### Step 3: Set up the variance equation
We know that the variance is given as 2. Thus, we can set up the equation:
\[ 2 = \frac{1}{n} \sum_{i=1}^{n} \left(X_i - \frac{n + 1}{2}\right)^2 \]

### Step 4: Expand the variance formula
The expression \( \sum_{i=1}^{n} \left(X_i - \mu\right)^2 \) can be expanded as follows:
\[ \sum_{i=1}^{n} \left(X_i - \frac{n + 1}{2}\right)^2 = \sum_{i=1}^{n} \left(X_i^2 - 2X_i\mu + \mu^2\right) \]
This can be simplified to:
\[ \sum_{i=1}^{n} X_i^2 - 2\mu \sum_{i=1}^{n} X_i + n\mu^2 \]

### Step 5: Calculate the necessary sums
1. The sum of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} X_i = \frac{n(n + 1)}{2} \]
2. The sum of the squares of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} X_i^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 6: Substitute into the variance equation
Substituting these sums into our variance equation gives:
\[ 2n = \frac{1}{n} \left( \frac{n(n + 1)(2n + 1)}{6} - 2 \cdot \frac{n + 1}{2} \cdot \frac{n(n + 1)}{2} + n \left(\frac{n + 1}{2}\right)^2 \right) \]

### Step 7: Simplify the equation
After simplifying, we find:
\[ 2n = \frac{1}{n} \left( \frac{n(n + 1)(2n + 1)}{6} - \frac{n(n + 1)^2}{2} + \frac{n(n + 1)^2}{4} \right) \]
Combining like terms leads to:
\[ 2n = \frac{n(n + 1)}{12} (2n + 1 - 6n + 3n + 3) \]

### Step 8: Solve for \( n \)
This leads to a quadratic equation in \( n \):
\[ n^2 - 1 = 24 \implies n^2 = 25 \implies n = 5 \text{ (since } n \text{ must be positive)} \]

### Final Answer
Thus, the value of \( n \) for which the numbers \( 1, 2, 3, \ldots, n \) have a variance of 2 is:
\[ \boxed{5} \]