What is the values of for which the numbers 1.2.3.n have variance 2

Options

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

Answer

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 defined as:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \]
where \( \mu \) is the mean of the numbers.

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

### Step 3: Substitute \( \mu \) into the variance formula
Now we need to calculate the variance using the formula:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} \left( i - \frac{n + 1}{2} \right)^2 \]
This expands to:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} \left( i^2 - (n + 1)i + \frac{(n + 1)^2}{4} \right) \]

### Step 4: Calculate the sum of squares
Using the formula for the sum of squares of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]
We can substitute this into our variance formula.

### Step 5: Set the variance equal to 2
We know that the variance is given as 2:
\[ \frac{1}{n} \left( \frac{n(n + 1)(2n + 1)}{6} - (n + 1) \frac{n(n + 1)}{2} + \frac{n(n + 1)^2}{4} \right) = 2 \]

### Step 6: Simplify the equation
Multiply both sides by \( n \) to eliminate the fraction:
\[ \frac{n(n + 1)(2n + 1)}{6} - (n + 1) \frac{n(n + 1)}{2} + \frac{n(n + 1)^2}{4} = 2n \]

### Step 7: Combine like terms
Rearranging and simplifying the terms will lead to a polynomial equation in \( n \).

### Step 8: Solve for \( n \)
After simplifying, we can solve the resulting polynomial equation for \( n \).

### Step 9: Check for valid solutions
Since \( n \) must be a positive integer, we will discard any negative solutions.

### Conclusion
After solving, we find that \( n = 5 \) satisfies the condition that the variance of the numbers \( 1, 2, 3, \ldots, n \) is equal to 2.