For which (k) is (k+7, 2k+3, 4k-5) an arithmetic progression?

Correct answer: 4

In an arithmetic progression, the differences between consecutive terms must be equal. Here, the first difference is \((2k+3)-(k+7)=k-4\), and the second difference is \((4k-5)-(2k+3)=2k-8\). Thus, \(k-4=2k-8\), which gives \(k=4\). Therefore, option C is correct. For instance, at \(k=3\), the two differences are not equal. Exam tip: For three terms to form an AP, equate the first and second consecutive differences.