To solve the problem, we need to understand the properties of an arithmetic progression (A.P.). An A.P. is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted as \(d\). ### Step-by-Step Explanation 1. **Understanding the Terms of an A.P.**: - The \(n\)-th term of an A.P. can be expressed using the formula: \[ a_n = a + (n-1)d \] where: - \(a\) is the first term, - \(d\) is the common difference, - \(n\) is the term number. 2. **Identifying the Sixth and Twelfth Terms**: - For the sixth term (\(a_6\)): \[ a_6 = a + (6-1)d = a + 5d \] - For the twelfth term (\(a_{12}\)): \[ a_{12} = a + (12-1)d = a + 11d \] 3. **Setting Up the Equation**: - According to the problem, the sixth term is half of the twelfth term: \[ a + 5d = \frac{1}{2}(a + 11d) \] 4. **Solving the Equation**: - To eliminate the fraction, multiply both sides by 2: \[ 2(a + 5d) = a + 11d \] - Expanding both sides gives: \[ 2a + 10d = a + 11d \] - Rearranging the equation to isolate \(a\): \[ 2a - a = 11d - 10d \] \[ a = d \] 5. **Conclusion**: - The first term \(a\) of the A.P. is equal to the common difference \(d\). Therefore, the correct answer is: \[ \text{Option D: the common difference} \] ### Explanation of Other Options - **Option A: Zero**: - If \(a = 0\), then \(d\) could be any value, but this does not satisfy the condition that \(a\) must equal \(d\). Thus, this option is incorrect. - **Option B: Half of the common difference**: - If \(a = \frac{1}{2}d\), then substituting this into our equation would not satisfy \(a = d\). Therefore, this option is also incorrect. - **Option C: Double the common difference**: - If \(a = 2d\), substituting this into our equation would again not satisfy \(a = d\). Hence, this option is incorrect as well. ### Summary of Key Points - The \(n\)-th term of an A.P. is given by \(a + (n-1)d\). - The relationship between the sixth and twelfth terms leads to the equation \(a + 5d = \frac{1}{2}(a + 11d)\). - Solving the equation reveals that the first term \(a\) is equal to the common difference \(d\). - The correct answer is that the first term of the A.P. is equal to the common difference (Option D). This thorough understanding of the properties of A.P.s and the algebraic manipulation involved is crucial for solving similar problems in the future.