The sum of the first five terms and the sum of the first ten terms of an AP are same. Which one of the following is the correct statement ?

Options

A. The first term must be negative
B. The common difference must be negative
C. Either the first term or the common difference is negative but not both
D. Both the first term and the common difference are negative

Explanation

To solve the problem, we need to analyze the conditions given in the question regarding the sums of the first five and ten terms of an arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding the Sum of Terms in an AP**:
The sum of the first \( n \) terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \left(2A + (n-1)D\right) \]
where \( A \) is the first term, \( D \) is the common difference, and \( n \) is the number of terms.

2. **Setting Up the Equations**:
For the first five terms (\( n = 5 \)):
\[ S_5 = \frac{5}{2} \left(2A + (5-1)D\right) = \frac{5}{2} \left(2A + 4D\right) \]
For the first ten terms (\( n = 10 \)):
\[ S_{10} = \frac{10}{2} \left(2A + (10-1)D\right) = 5 \left(2A + 9D\right) \]

3. **Setting the Sums Equal**:
According to the problem, the sum of the first five terms is equal to the sum of the first ten terms:
\[ \frac{5}{2} (2A + 4D) = 5 (2A + 9D) \]

4. **Simplifying the Equation**:
Multiply both sides by 2 to eliminate the fraction:
\[ 5(2A + 4D) = 10(2A + 9D) \]
Expanding both sides:
\[ 10A + 20D = 20A + 90D \]

5. **Rearranging the Terms**:
Move all terms involving \( A \) to one side and all terms involving \( D \) to the other side:
\[ 10A - 20A = 90D - 20D \]
This simplifies to:
\[ -10A = 70D \]
Dividing both sides by -10 gives:
\[ A = -7D \]

6. **Analyzing the Result**:
The equation \( A = -7D \) indicates that either \( A \) or \( D \) must be negative. If \( D \) is positive, \( A \) will be negative, and if \( D \) is negative, \( A \) will be positive. Therefore, both cannot be negative simultaneously.

### Conclusion:
The correct statement is that either the first term \( A \) is negative or the common difference \( D \) is negative, but not both. Thus, the answer is:

**Option 3: Either the first term or the common difference is negative but not both.**