In the number series 10, 29, 66, 127, 218, the next term is _______. – UGC NET Question

The series can be described by the formula (a_n = n^3 + 2) for n starting from 2. For n=2,3,4,5 and 6 the values are 8+2=10, 27+2=29, 64+2=66, 125+2=127 and 216+2=218, matching the given terms. The next index is n=7, giving (7^3+2 = 343+2 = 345). Thus 345 is the natural continuation of this cubic-based sequence.

Option A:
Option A, 337, does not arise from the expression (n^3+2) for any integer n that continues the index sequence. Using 337 would require abandoning the clear functional rule that explains all earlier terms. Therefore 337 is not a valid next term.

Option B:
Option B equals 345, which is exactly (7^3+2), following the same formula used to generate 10, 29, 66, 127 and 218. Because the rule holds for every term including the new one, 345 is the correct answer.

Option C:
Option C, 349, is larger than the value predicted by the cubic formula and cannot be justified without changing the rule. It does not match (n^3+2) for any integer n that logically follows. Hence 349 is not consistent with the pattern.

Option D:
Option D, 353, similarly fails to satisfy the relation (a_n = n^3+2) for the next integer in sequence. Adopting 353 would make the series lose its simple and elegant algebraic structure. Thus 353 is not the right continuation.