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 terms 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

Answer

To solve the problem, we need to find the relationship between the first term (a) and the common difference (d) of an arithmetic progression (AP) given that the sum of the first five terms is equal to the sum of the first ten terms.

### Step-by-Step Solution:

1. **Understand the formula for the sum of the first n terms of an AP:**
The sum of the first n terms (S_n) of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

2. **Calculate the sum of the first 5 terms (S_5):**
Using the formula for \( n = 5 \):
\[ S_5 = \frac{5}{2} \times (2a + (5-1)d) = \frac{5}{2} \times (2a + 4d) = \frac{5}{2} \times (2a + 4d) \]

3. **Calculate the sum of the first 10 terms (S_10):**
Using the formula for \( n = 10 \):
\[ S_{10} = \frac{10}{2} \times (2a + (10-1)d) = 5 \times (2a + 9d) = 5 \times (2a + 9d) \]

4. **Set the two sums equal to each other:**
Since we know that \( S_5 = S_{10} \):
\[ \frac{5}{2} \times (2a + 4d) = 5 \times (2a + 9d) \]

5. **Simplify the equation:**
Dividing both sides by 5:
\[ \frac{1}{2} \times (2a + 4d) = 2a + 9d \]
Multiply through by 2 to eliminate the fraction:
\[ 2a + 4d = 4a + 18d \]

6. **Rearranging the equation:**
Bringing all terms involving \( a \) to one side and terms involving \( d \) to the other:
\[ 2a - 4a = 18d - 4d \]
This simplifies to:
\[ -2a = 14d \]
Dividing both sides by -2 gives:
\[ a = -7d \]

7. **Conclusion:**
The relationship derived indicates that either the first term \( a \) is negative or the common difference \( d \) is negative, but not both. Thus, the correct statement is:
- Either the first term or the common difference is negative, but not both.