To solve the problem, we need to analyze the information given about the arithmetic progression (AP) and derive the common difference. Let's break it down step-by-step. ### Step 1: Understanding the Terms of an AP In an arithmetic progression, each term can be expressed in terms of the first term \( a \) and the common difference \( d \). The \( n \)-th term of an AP can be given by the formula: \[ T_n = a + (n-1)d \] ### Step 2: Expressing the Given Terms From the problem, we know: - The 7th term \( T_7 \) is given by: \[ T_7 = a + 6d \] - The 3rd term \( T_3 \) is given by: \[ T_3 = a + 2d \] According to the problem, the 7th term is twice the 3rd term: \[ T_7 = 2 \cdot T_3 \] Substituting the expressions for \( T_7 \) and \( T_3 \): \[ a + 6d = 2(a + 2d) \] ### Step 3: Simplifying the Equation Now, let's simplify the equation: \[ a + 6d = 2a + 4d \] Rearranging gives: \[ 6d - 4d = 2a - a \] \[ 2d = a \] This tells us that the first term \( a \) is equal to \( 2d \). ### Step 4: Using the Sum of the First Four Terms Next, we know that the sum of the first four terms is 42. The sum of the first \( n \) terms of an AP can be calculated using the formula: \[ S_n = \frac{n}{2} \times (2a + (n-1)d) \] For the first four terms (\( n = 4 \)): \[ S_4 = \frac{4}{2} \times (2a + 3d) = 2(2a + 3d) \] Setting this equal to 42: \[ 2(2a + 3d) = 42 \] Dividing both sides by 2: \[ 2a + 3d = 21 \] ### Step 5: Substituting \( a \) in Terms of \( d \) Now, we substitute \( a = 2d \) into the equation: \[ 2(2d) + 3d = 21 \] This simplifies to: \[ 4d + 3d = 21 \] \[ 7d = 21 \] Dividing both sides by 7 gives: \[ d = 3 \] ### Conclusion: Finding the Common Difference Thus, the common difference \( d \) is 3. ### Step 6: Verifying the Other Options Now, let's check the other options to confirm that they are incorrect: - **Option A: 6** - If \( d = 6 \), then \( a = 2d = 12 \). The sum of the first four terms would be \( 2(2a + 3d) = 2(24 + 18) = 84 \), which is not 42. - **Option C: 2** - If \( d = 2 \), then \( a = 4 \). The sum would be \( 2(2a + 3d) = 2(8 + 6) = 28 \), which is not 42. - **Option D: 1** - If \( d = 1 \), then \( a = 2 \). The sum would be \( 2(2a + 3d) = 2(4 + 3) = 14 \), which is not 42. ### Revision Summary - The 7th term of an AP can be expressed as \( T_7 = a + 6d \) and the 3rd term as \( T_3 = a + 2d \). - The relationship \( T_7 = 2T_3 \) leads to \( a = 2d \). - The sum of the first four terms gives a second equation \( 2a + 3d = 21 \). - Solving these equations reveals that the common difference \( d \) is 3. The correct answer is **B. 3**.