If the mean of the squares of first n natural numbers is 105 , then find the median of the first n natural numbers .

Options

A. 8
B. 9
C. 10
D. 11

Explanation

To solve the problem step by step, we need to find the median of the first n natural numbers given that the mean of their squares is 105.

### Step 1: Understand the Mean of Squares
The mean of the squares of the first n natural numbers is given by the formula:

\[ \text{Mean} = \frac{\sum_{k=1}^{n} k^2}{n} \]

### Step 2: Use the Formula for Sum of Squares
The sum of the squares of the first n natural numbers is given by:

\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 3: Set Up the Equation
According to the problem, the mean of the squares is 105. Therefore, we can set up the equation:

\[ \frac{\frac{n(n + 1)(2n + 1)}{6}}{n} = 105 \]

This simplifies to:

\[ \frac{(n + 1)(2n + 1)}{6} = 105 \]

### Step 4: Multiply Both Sides by 6
To eliminate the fraction, multiply both sides by 6:

\[ (n + 1)(2n + 1) = 630 \]

### Step 5: Expand the Left Side
Expanding the left side gives:

\[ 2n^2 + 3n + 1 = 630 \]

### Step 6: Rearrange the Equation
Rearranging the equation to set it to zero:

\[ 2n^2 + 3n + 1 - 630 = 0 \]

This simplifies to:

\[ 2n^2 + 3n - 629 = 0 \]

### Step 7: Solve the Quadratic Equation
Now, we can solve the quadratic equation using the quadratic formula:

\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \(a = 2\), \(b = 3\), and \(c = -629\).

Calculating the discriminant:

\[ b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot (-629) = 9 + 5024 = 5033 \]

Now, substituting back into the quadratic formula:

\[ n = \frac{-3 \pm \sqrt{5033}}{4} \]

Calculating \(\sqrt{5033}\) gives approximately 71.0.

Thus:

\[ n = \frac{-3 + 71}{4} \quad \text{(only the positive root is valid)} \]

Calculating this gives:

\[ n = \frac{68}{4} = 17 \]

### Step 8: Find the Median
Since \(n = 17\), the first 17 natural numbers are \(1, 2, 3, \ldots, 17\).

For an odd number of terms, the median is the middle term:

\[ \text{Median} = \frac{n + 1}{2} = \frac{17 + 1}{2} = 9 \]

### Final Answer
Thus, the median of the first n natural numbers is:

\[ \boxed{9} \]