To solve the problem, we need to understand the properties of an arithmetic progression (A.P.). In an A.P., each term can be expressed in terms of the first term \( a \) and the common difference \( d \). ### Step-by-Step Explanation 1. **Understanding the Terms of A.P.**: - The \( n \)-th term of an A.P. can be calculated using the formula: \[ T_n = a + (n-1)d \] - Here, \( T_n \) is the \( n \)-th term, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number. 2. **Finding the 9th and 5th Terms**: - For the 9th term (\( T_9 \)): \[ T_9 = a + (9-1)d = a + 8d \] - For the 5th term (\( T_5 \)): \[ T_5 = a + (5-1)d = a + 4d \] 3. **Setting Up the Equation**: - According to the problem, the 9th term is five times the 5th term: \[ T_9 = 5 \times T_5 \] - Substituting the expressions we found: \[ a + 8d = 5(a + 4d) \] 4. **Expanding and Rearranging**: - Expanding the right side: \[ a + 8d = 5a + 20d \] - Now, rearranging the equation to isolate terms involving \( a \) and \( d \): \[ a + 8d - 5a - 20d = 0 \] \[ -4a - 12d = 0 \] - Dividing the entire equation by -4 gives: \[ a + 3d = 0 \] 5. **Conclusion**: - This means that the relationship between \( a \) and \( d \) is: \[ a + 3d = 0 \] - Therefore, the correct option is **C**. ### Why Other Options Are Incorrect - **Option A: \( 2a + 2 = 0 \)**: - This equation does not relate \( a \) and \( d \) in the context of the problem. It suggests a specific value for \( a \) but does not involve \( d \) at all. - **Option B: \( 3a + 5d = 0 \)**: - This equation does not match our derived relationship. It suggests a different linear relationship between \( a \) and \( d \) that is not supported by the problem's conditions. - **Option D: \( a + 2d = 0 \)**: - This option suggests a different relationship that does not satisfy the condition given in the problem. It implies a different ratio between \( a \) and \( d \) than what we derived. ### Revision Summary - The \( n \)-th term of an A.P. is given by \( T_n = a + (n-1)d \). - The relationship derived from the condition that the 9th term is five times the 5th term is \( a + 3d = 0 \). - The correct answer is option **C**. - Other options do not correctly represent the relationship between \( a \) and \( d \) based on the problem's conditions.