Let `x_(1),x_(2),…x_(n)` be n observations such that `Sigmax_(i)^(2)=500 and Sigmax_(1)=100`. Then, an impossible value of n among the following is

Options

A. 24
B. 18
C. 29
D. 22

Answer

To solve the problem, we need to analyze the given conditions and apply the concept of the Cauchy-Schwarz inequality.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have \( \Sigma x_i^2 = 500 \) (the sum of squares of the observations).
- We have \( \Sigma x_i = 100 \) (the sum of the observations).

2. **Applying the Cauchy-Schwarz Inequality**:
- According to the Cauchy-Schwarz inequality, for any real numbers \( x_1, x_2, \ldots, x_n \):
\[ (\Sigma x_i^2)(\Sigma 1) \geq (\Sigma x_i)^2 \]
- Here, \( \Sigma 1 = n \) (since we have n observations).
- Therefore, we can write:
\[ (\Sigma x_i^2)(n) \geq (\Sigma x_i)^2 \]
- Substituting the known values:
\[ 500n \geq (100)^2 \]
- This simplifies to:
\[ 500n \geq 10000 \]

3. **Solving for n**:
- Dividing both sides by 500:
\[ n \geq \frac{10000}{500} \]
- Simplifying this gives:
\[ n \geq 20 \]

4. **Identifying Impossible Values**:
- The question asks for an impossible value of \( n \). Since we found that \( n \) must be at least 20, any value of \( n \) that is less than 20 is impossible.
- Therefore, if the options include values such as 18, that would be impossible.

### Conclusion:
The impossible value of \( n \) among the given options is **18**.