In the number series 3, 30, 115, 318, 723, the next term is _______. – UGC NET Question

The series follows the rule (a_n = n^4 + 4n^2 - 2) with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 3, 30, 115, 318 and 723, as in the question. For n = 6 we compute (6^4 + 4×6^2 - 2 = 1296 + 144 - 2 = 1438). Therefore 1438 is the correct next term in the sequence.

Option A:
Option A, 1414, is 24 less than the formula’s output and does not equal (6^4 + 4×6^2 - 2). It would impose an artificial reduction that the pattern does not support. Hence 1414 is not consistent with the rule.

Option B:
Option B, 1438, matches exactly the result of the expression for n = 6. It preserves the quartic and quadratic contributions in exactly the same way as for earlier terms. This alignment makes 1438 the correct answer.

Option C:
Option C, 1454, overshoots the computed value and cannot be derived from the formula for n = 6. Choosing 1454 would disrupt the precise algebraic structure of the series. Thus 1454 is not valid.

Option D:
Option D, 1470, deviates even more from 1438 and once again fails to satisfy the rule. Using 1470 would break the tight connection between index and term, so it is not correct.