In the number series 6, 25, 62, 123, 214, the next term is _______. – UGC NET Question

The given series is generated by the rule (a_n = n^3 - 2) with n starting from 2. For n=2 we get 2³−2 = 8−2 = 6, for n=3 we get 27−2 = 25, for n=4 we obtain 64−2 = 62, for n=5 we obtain 125−2 = 123 and for n=6 we obtain 216−2 = 214. The next index is n=7, giving 7³−2 = 343−2 = 341 as the next term. Hence 341 is the correct continuation of the sequence.

Option A:
Option A, 333, does not equal (n^3-2) for n=7 or for any integer n that follows in the natural order of indices. Using 333 would require modifying the clear cubic rule that currently explains every term. Thus 333 is not consistent with the pattern.

Option B:
Option B, 339, is also not of the form (7^3-2) and does not result from the same functional relation. It introduces an arbitrary deviation from the established cubic structure. Hence 339 is not a valid next term.

Option C:
Option C equals 341, which is precisely 7³−2 and continues the rule (a_n = n^3-2) seamlessly. Since this one formula accounts for all given terms and the candidate term, 341 correctly extends the series.

Option D:
Option D, 347, is larger than the value predicted by 7³−2 and fails to satisfy the same polynomial relation. Selecting 347 would break the tight algebraic pattern that unifies the sequence. Therefore 347 cannot be accepted as the next term.