The arithmetic mean of first `n` natural numbers, is

Options

A. `(pi)/(2)`
B. `(n+1)/(2)`
C. `(n(n+1))/(2)`
D. `(n-1)/(2)`

Explanation

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

### Step 1: Understand the formula for arithmetic mean
The arithmetic mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} \]

### Step 2: Calculate the sum of the first `n` natural numbers
The sum of the first `n` natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]

### Step 3: Determine the total number of observations
For the first `n` natural numbers, the total number of observations is simply `n`.

### Step 4: Substitute the values into the mean formula
Now, we substitute the sum and the total number of observations into the mean formula:
\[ \text{Mean} = \frac{\frac{n(n + 1)}{2}}{n} \]

### Step 5: Simplify the expression
We can simplify the expression:
\[ \text{Mean} = \frac{n(n + 1)}{2n} \]
Here, the `n` in the numerator and denominator cancels out:
\[ \text{Mean} = \frac{n + 1}{2} \]

### Conclusion
Thus, the arithmetic mean of the first `n` natural numbers is:
\[ \text{Mean} = \frac{n + 1}{2} \]

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