The arithmetic mean of the squares of first n natural numbers is

Options

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

Answer

To find the arithmetic mean of the squares of the first n natural numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Arithmetic Mean**:
The arithmetic mean (average) is calculated by taking the sum of all data points and dividing it by the total number of data points.

2. **Identify the Data Points**:
The first n natural numbers are 1, 2, 3, ..., n. We need to find the squares of these numbers, which are \(1^2, 2^2, 3^2, \ldots, n^2\).

3. **Calculate the Sum of Squares**:
The sum of the squares of the first n natural numbers can be expressed using the formula:
\[ \text{Sum of squares} = 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \]

4. **Determine the Total Number of Data Points**:
The total number of data points is simply n, since we are considering the first n natural numbers.

5. **Calculate the Arithmetic Mean**:
Now, we can calculate the arithmetic mean of the squares:
\[ \text{Arithmetic Mean} = \frac{\text{Sum of squares}}{\text{Total number of data points}} = \frac{\frac{n(n + 1)(2n + 1)}{6}}{n} \]

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

### Final Answer:
The arithmetic mean of the squares of the first n natural numbers is:
\[ \frac{(n + 1)(2n + 1)}{6} \]