In the number series 3, 18, 57, 132, 255, the next term is _______. – UGC NET Question

This sequence is described by the formula (a_n = 2n^3 + n) with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 3, 18, 57, 132 and 255, matching the given series exactly. For n = 6 the expression gives (2×6^3 + 6 = 432 + 6 = 438). Thus 438 is the only value that maintains this cubic-plus-linear rule.

Option A:
Option A, 430, is smaller than the computed value and does not equal (2×6^3 + 6). It would require changing the formula at the final term, which has no support from the preceding structure. Therefore 430 is not a valid continuation.

Option B:
Option B, 438, matches precisely the value produced by (a_n = 2n^3 + n) when n = 6. It preserves the same relationship between position and value that explains all earlier terms. For this reason, 438 is the correct next term in the series.

Option C:
Option C, 442, overshoots the cubic prediction and is not the output of the given expression for any suitable n after 5. Adopting 442 would arbitrarily inflate the term and break the algebraic pattern. Hence 442 cannot be accepted.

Option D:
Option D, 454, deviates even more from the expected value and similarly fails to satisfy the rule (2n^3 + n) for n = 6. Selecting 454 would destroy the clean cubic structure of the sequence, so it is not correct.