In the number series 4, 33, 162, 511, 1248, the next term is _______. – UGC NET Question

This series is generated by the expression (a_n = 2n^4 - n + 3) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2−1+3 = 4, 32−2+3 = 33, 162−3+3 = 162, 512−4+3 = 511 and 1250−5+3 = 1248. For n = 6 we compute (2×6^4 - 6 + 3 = 2592 - 3 = 2589). Therefore 2589 is the correct next term.

Option A:
Option A, 2589, coincides with the value of (2n^4 - n + 3) when n = 6. It maintains the same combination of quartic and linear terms that explain all previous numbers. Because it fits the rule exactly, 2589 is the correct continuation.

Option B:
Option B, 2557, is 32 less than the computed value and does not satisfy the expression for n = 6. It would require a sudden drop that has no basis in the pattern, so 2557 is not valid.

Option C:
Option C, 2613, overshoots the predicted value and cannot be obtained from the generating rule. Selecting 2613 would arbitrarily increase the last term and break the algebraic structure. Thus 2613 is not correct.

Option D:
Option D, 2645, deviates even further from 2589 and is not the output of (2n^4 - n + 3) for any integer n continuing the sequence. Adopting 2645 would disrupt the pattern, so it is not the right answer.