The mean of the first n natural numbers is 10, then the number of observations is__________

Answer

To find the number of observations (n) when the mean of the first n natural numbers is given as 10, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for mean**:
The mean (average) of a set of numbers is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

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

3. **Set up the equation using the given mean**:
We know from the problem that the mean is 10. Therefore, we can set up the equation:
\[ 10 = \frac{\frac{n(n + 1)}{2}}{n} \]

4. **Simplify the equation**:
When we simplify the right side of the equation:
\[ 10 = \frac{n(n + 1)}{2n} \]
This simplifies to:
\[ 10 = \frac{n + 1}{2} \]

5. **Multiply both sides by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ 20 = n + 1 \]

6. **Solve for n**:
Now, subtract 1 from both sides to find n:
\[ n = 20 - 1 = 19 \]

7. **Conclusion**:
Therefore, the number of observations (n) is:
\[ \boxed{19} \]