For which (n) will (n+2, 2n+5, 4n+8) form an arithmetic progression?

Correct answer: \(0\)

In an arithmetic progression, the differences between consecutive terms must be equal. Here, the difference between the second and first terms is \((2n+5)-(n+2)=n+3\), while the difference between the third and second terms is \((4n+8)-(2n+5)=2n+3\). Thus, \(n+3=2n+3\), giving \(n=0\). On substituting \(n=0\), the terms are \(2,5,8\), with common difference \(3\). For example, at \(n=1\), the differences are \(4\) and \(5\), so it is not an AP. Exam tip: For three terms to form an AP, equate the two consecutive differences.