Why is the sequence (2,6,12,20,\ldots) not an arithmetic progression?

Correct answer: Because consecutive differences are not equal

The direct answer is option D: consecutive differences are not equal. An arithmetic progression requires one fixed common difference between every two consecutive terms. For the sequence 2, 6, 12, 20, calculate carefully: \(6-2=4\), \(12-6=6\), and \(20-12=8\). The differences are 4, 6, and 8, so they change rather than remain constant. Therefore the sequence is not an AP. Option A is wrong because the first term being 2 has no bearing on whether the differences are equal; an AP may start with any number. Option B is wrong because increasing terms alone do not guarantee an AP. For example, 1, 2, 4, 8 is increasing but its differences are 1, 2, 4, not equal. Option C is wrong because being positive also does not decide the type of sequence; an AP can contain positive, negative, or zero terms. Option D states the exact defining failure: the consecutive differences are unequal. The terms actually follow another pattern, since they can be written as products of consecutive integers, but that is not needed to answer this question. The key test is always subtraction of neighboring terms. Memory cue: AP means equal differences, not merely increasing terms or positive terms.