The sequence (2, 2+k, 2+4k, 2+9k) can be an arithmetic progression. Under what condition?

Correct answer: \(k=0\)

In an arithmetic progression, the differences between consecutive terms must be equal. Here the differences are \(k\), \(3k\), and \(5k\). For them to be equal, \(k=3k=5k\), which is possible only when \(k=0\). Then the sequence becomes \(2,2,2,2\), with common difference \(0\). For example, when \(k=1\), the differences are \(1,3,5\), so it is not an AP. Exam tip: To test an AP, compare all consecutive differences.