The mean of the squares of the numbers 1,2,3,4, `cdots` n is

Options

A. `(1)/(6) (n+1) (2n+1)`
B. `(1)/(6) (n+1)`
C. `(1)/(6) (2n+1)`
D. None of these

Answer

To find the mean of the squares of the numbers from 1 to \( n \), we can follow these steps:

### Step 1: Understand the Mean of Squares
The mean of squares of the first \( n \) natural numbers is calculated by taking the sum of the squares of these numbers and dividing it by \( n \).

### Step 2: Calculate the Sum of Squares
The formula for the sum of squares of the first \( n \) natural numbers is given by:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 3: Set Up the Mean Formula
The mean of the squares can be expressed as:
\[ \text{Mean} = \frac{\text{Sum of squares}}{n} = \frac{\sum_{i=1}^{n} i^2}{n} \]

### Step 4: Substitute the Sum of Squares Formula
Now, substituting the formula for the sum of squares into the mean formula, we get:
\[ \text{Mean} = \frac{\frac{n(n + 1)(2n + 1)}{6}}{n} \]

### Step 5: Simplify the Expression
When we simplify the expression, we can cancel \( n \) in the numerator and denominator:
\[ \text{Mean} = \frac{(n + 1)(2n + 1)}{6} \]

### Final Answer
Thus, the mean of the squares of the numbers from 1 to \( n \) is:
\[ \text{Mean} = \frac{(n + 1)(2n + 1)}{6} \]

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