2) If \( k+1,2 k-1 \& 3 k+1 \) are three conseactive terme of the possible values of the common dufference

To find the possible values of the common difference for the three consecutive terms \( k+1 \), \( 2k-1 \), and \( 3k+1 \), we need to set up the condition for an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we can express this condition mathematically as follows: 1. The difference between the second term and the first term should equal the difference between the third term and the second term: \[ (2k - 1) - (k + 1) = (3k + 1) - (2k - 1) \] Now, let's simplify both sides of the equation step by step. ### Step 1: Simplify the left side \[ (2k - 1) - (k + 1) = 2k - 1 - k - 1 = k - 2 \] ### Step 2: Simplify the right side \[ (3k + 1) - (2k - 1) = 3k + 1 - 2k + 1 = k + 2 \] ### Step 3: Set the two sides equal Now we have: \[ k - 2 = k + 2 \] ### Step 4: Solve for \( k \) Subtract \( k \) from both sides: \[ -2 = 2 \] This is a contradiction, which means there are no values of \( k \) that satisfy the condition for these three terms to be in an arithmetic sequence. ### Conclusion The possible values of the common difference do not exist for the terms \( k+1 \), \( 2k-1 \), and \( 3k+1 \) as they cannot form an arithmetic sequence.