In the number series 2, 9, 28, 65, 126, the next term is _______. – UGC NET Question

Each term in the series can be expressed as n³ + 1 for successive integers n. When n = 1, 2, 3, 4 and 5, we obtain 1³ + 1 = 2, 2³ + 1 = 9, 3³ + 1 = 28, 4³ + 1 = 65 and 5³ + 1 = 126. The next integer is 6, so the next term should be 6³ + 1. This equals 216 + 1 = 217, which perfectly matches the observed rule.

Option A:
Option A gives 217, which fits the formula n³ + 1 when n = 6. Extending the sequence to 2, 9, 28, 65, 126, 217 keeps the entire series under one simple cubic rule. This confirms that 217 is the correct next term.

Option B:
Option B suggests 218, which is one more than 217 and does not correspond to n³ + 1 for any integer in the natural continuation. It breaks the tight relationship between n and the terms. Therefore, 218 cannot be accepted as correct.

Option C:
Option C offers 225, which is a square number (15²) but not of the form n³ + 1 in this context. It introduces a different kind of pattern unrelated to the existing terms. Hence, 225 does not match the logic of the series.

Option D:
Option D provides 230, which likewise fails to satisfy the n³ + 1 relation for any integer following 5 in sequence. Choosing 230 would abandon the clearly established cubic structure. Thus, 230 is not the right answer.