The mean of the first n natural numbers is 32, find n.

Options

A. 64
B. 63
C. 65
D. 62

Answer

To find the value of \( n \) when the mean of the first \( n \) natural numbers is 32, we can follow these steps:

### Step 1: Understand the formula for the mean
The mean of a set of numbers is calculated as the sum of the numbers divided by the count of the numbers. For the first \( n \) natural numbers, the mean can be expressed as:
\[ \text{Mean} = \frac{\text{Sum of first } n \text{ natural numbers}}{n} \]

### Step 2: Calculate the sum of the first \( n \) natural numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]

### Step 3: Set up the equation for the mean
According to the problem, the mean is 32. Therefore, we can set up the equation:
\[ \frac{\frac{n(n + 1)}{2}}{n} = 32 \]

### Step 4: Simplify the equation
By simplifying the left side, we get:
\[ \frac{n(n + 1)}{2n} = \frac{n + 1}{2} \]
So, the equation becomes:
\[ \frac{n + 1}{2} = 32 \]

### Step 5: Cross-multiply to eliminate the fraction
To eliminate the fraction, we can cross-multiply:
\[ n + 1 = 2 \times 32 \]
This simplifies to:
\[ n + 1 = 64 \]

### Step 6: Solve for \( n \)
Now, we can solve for \( n \) by isolating it:
\[ n = 64 - 1 \]
Thus, we find:
\[ n = 63 \]

### Final Answer
The value of \( n \) is \( 63 \).

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