Let `x_(1), x_(2),….,x_(n)` be n observation such that `sum(x_(i))^(2)=400 and sumx_(i)=40`, then a possible value of n among the following is

Options

A. 5
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the given conditions and apply the Cauchy-Schwarz inequality (or the property of arithmetic mean and quadratic mean) to find a suitable value for \( n \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have \( \sum_{i=1}^{n} x_i^2 = 400 \) (the sum of squares of the observations).
- We also have \( \sum_{i=1}^{n} x_i = 40 \) (the sum of the observations).

2. **Using the Cauchy-Schwarz Inequality**:
- According to the Cauchy-Schwarz inequality, we have:
\[ \left( \sum_{i=1}^{n} x_i^2 \right) \left( \sum_{i=1}^{n} 1 \right) \geq \left( \sum_{i=1}^{n} x_i \right)^2 \]
- Here, \( \sum_{i=1}^{n} 1 = n \), so we can rewrite the inequality as:
\[ \left( \sum_{i=1}^{n} x_i^2 \right) n \geq \left( \sum_{i=1}^{n} x_i \right)^2 \]

3. **Substituting the Given Values**:
- Substitute the known values into the inequality:
\[ 400n \geq 40^2 \]
- Calculate \( 40^2 \):
\[ 40^2 = 1600 \]
- So, we have:
\[ 400n \geq 1600 \]

4. **Solving for \( n \)**:
- Divide both sides by 400:
\[ n \geq \frac{1600}{400} \]
- Simplifying gives:
\[ n \geq 4 \]

5. **Finding Possible Values of \( n \)**:
- The question asks for a possible value of \( n \) among the options provided. Since we found that \( n \) must be greater than or equal to 4, we check the options:
- If the options are 1, 2, 3, and 5, then the only suitable value is:
\[ n = 5 \]

### Conclusion:
Thus, a possible value of \( n \) is **5**.