If the arithmetic mean of the first n natural numbers is 15 , then n is `"__________"` .

Options

A. 15
B. 20
C. 14
D. 29

Explanation

To find the value of \( n \) such that the arithmetic mean of the first \( n \) natural numbers is 15, we can follow these steps:

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

### Step 2: Identify the sum of the first \( n \) natural numbers
The first \( n \) natural numbers are \( 1, 2, 3, \ldots, n \). The sum of these numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]

### Step 3: Set up the equation for the mean
Given that the arithmetic mean is 15, we can set up the equation:
\[ 15 = \frac{\text{Sum of first } n \text{ natural numbers}}{n} \]
Substituting the sum from Step 2 into this equation, we have:
\[ 15 = \frac{\frac{n(n + 1)}{2}}{n} \]

### Step 4: Simplify the equation
We can simplify the equation:
\[ 15 = \frac{n(n + 1)}{2n} \]
This simplifies to:
\[ 15 = \frac{n + 1}{2} \]

### Step 5: Solve for \( n \)
To eliminate the fraction, multiply both sides by 2:
\[ 30 = n + 1 \]
Now, subtract 1 from both sides:
\[ n = 30 - 1 \]
Thus, we find:
\[ n = 29 \]

### Conclusion
The value of \( n \) is \( 29 \).

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