What is the Arithmetic mean of the first'n' natural numbers ?

Options

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

Answer

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

### Step 1: Understand the concept of natural numbers
Natural numbers are the positive integers starting from 1, i.e., 1, 2, 3, ..., n.

### Step 2: Calculate the sum of the first 'n' natural numbers
The formula for the sum of the first 'n' natural numbers is:
\[ S = \frac{n(n + 1)}{2} \]
where \( S \) is the sum.

### Step 3: Determine the number of terms
The number of terms in the first 'n' natural numbers is simply \( n \).

### Step 4: Calculate the arithmetic mean
The arithmetic mean (AM) is calculated using the formula:
\[ \text{AM} = \frac{\text{Sum of the numbers}}{\text{Number of terms}} \]
Substituting the values we have:
\[ \text{AM} = \frac{S}{n} = \frac{\frac{n(n + 1)}{2}}{n} \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ \text{AM} = \frac{n(n + 1)}{2n} = \frac{n + 1}{2} \]

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

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