In the number series 2, 14, 74, 242, 602, the next term is _______. – UGC NET Question

The sequence is generated by the rule aₙ = n⁴−n²+2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1−1+2 = 2, 16−4+2 = 14, 81−9+2 = 74, 256−16+2 = 242 and 625−25+2 = 602. For n = 6 the expression gives 1296−36+2 = 1262. Hence 1262 is the term that preserves this quartic-minus-square pattern.

Option A:
Option A, 1222, is 40 less than the value predicted by n⁴−n²+2 for n = 6. It does not satisfy the formula and would require adjusting the result in a way that is not supported by the earlier data. Therefore 1222 is not the correct continuation.

Option B:
Option B, 1238, also fails to match 1296−36+2 and breaks the tight link between index and term. It represents a partial correction but still does not arise from the generating rule. Thus 1238 cannot be accepted.

Option C:
Option C, 1292, overshoots the correct value and cannot be written as n⁴−n²+2 for n = 6. Adopting 1292 would abandon the exact quartic pattern that fits the sequence so well. Hence it is not valid.

Option D:
Option D, 1262, exactly equals 6⁴−6²+2 and therefore continues the same polynomial relationship. It keeps the structure of the series fully consistent from the first term to the next one. For this reason, 1262 is the correct next term.