Let `A_(1), A_(2)…….A_(7)` be a polygon and `a(1), a_(2)……a_(7)` be the complex numbers representing vertices `A_(1), A_(2)……A_(7)`. If `|a_(1)|=|a_(2)|=……….|a_(7)=R`, then `sum_(1le i lt j le 7)|a_(i)+a_(j)|^(2)`

Options

A. greater than `30R^(2)`
B. has minimum value as `35R^(2)`
C. has its minimum value in `(25R^(2),45R^(2))`
D. is less than `45R^(2)`

Explanation

To solve the problem, we need to evaluate the expression:

\[ \sum_{1 \leq i < j \leq 7} |a_i + a_j|^2 \]

Given that \(|a_1| = |a_2| = \ldots = |a_7| = R\), we can denote \( |a_i|^2 = R^2 \) for all \( i \).

### Step 1: Expand the expression
We start by expanding the expression \( |a_i + a_j|^2 \):

\[ |a_i + a_j|^2 = (a_i + a_j)(\overline{a_i + a_j}) = |a_i|^2 + |a_j|^2 + a_i \overline{a_j} + \overline{a_i} a_j \]

### Step 2: Substitute the magnitudes
Since \(|a_i|^2 = R^2\) for all \(i\):

\[ |a_i + a_j|^2 = R^2 + R^2 + a_i \overline{a_j} + \overline{a_i} a_j = 2R^2 + a_i \overline{a_j} + \overline{a_i} a_j \]

### Step 3: Sum over all pairs
Now we need to sum this expression over all pairs \( (i, j) \) where \( 1 \leq i < j \leq 7 \):

\[ \sum_{1 \leq i < j \leq 7} |a_i + a_j|^2 = \sum_{1 \leq i < j \leq 7} \left( 2R^2 + a_i \overline{a_j} + \overline{a_i} a_j \right) \]

### Step 4: Calculate the constant term
The term \( 2R^2 \) contributes:

\[ \sum_{1 \leq i < j \leq 7} 2R^2 = 2R^2 \cdot \binom{7}{2} = 2R^2 \cdot 21 = 42R^2 \]

### Step 5: Calculate the cross terms
Next, we need to evaluate the sum of the cross terms:

\[ \sum_{1 \leq i < j \leq 7} (a_i \overline{a_j} + \overline{a_i} a_j) \]

This can be rewritten as:

\[ \sum_{i=1}^{7} \sum_{j=1}^{7} a_i \overline{a_j} - \sum_{i=1}^{7} |a_i|^2 \]

The first term sums over all pairs, which is \( \left( \sum_{i=1}^{7} a_i \right) \left( \sum_{j=1}^{7} \overline{a_j} \right) \). Let \( S = \sum_{i=1}^{7} a_i \):

\[ = |S|^2 - 7R^2 \]

### Step 6: Combine the results
Putting everything together, we have:

\[ \sum_{1 \leq i < j \leq 7} |a_i + a_j|^2 = 42R^2 + |S|^2 - 7R^2 = 35R^2 + |S|^2 \]

### Step 7: Analyze the result
Since \( |S|^2 \geq 0 \), we conclude:

\[ \sum_{1 \leq i < j \leq 7} |a_i + a_j|^2 \geq 35R^2 \]

### Conclusion
Thus, the minimum value of the sum is \( 35R^2 \). The options provided indicate that the sum can be greater than \( 30R^2 \), equal to \( 35R^2 \), and less than \( 45R^2 \). Therefore, the correct answer is:

- Greater than \( 30R^2 \)
- Minimum value is \( 35R^2 \)
- Less than \( 45R^2 \) cannot be determined without additional information.