In the number series 4, 54, 258, 796, 1920, the next term is _______. – UGC NET Question

The pattern here is given by aₙ = 3n⁴ + 2n² − n, with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3·1⁴+2·1²−1 = 4, 3·2⁴+2·2²−2 = 54, 3·3⁴+2·3²−3 = 258, 3·4⁴+2·4²−4 = 796 and 3·5⁴+2·5²−5 = 1920, which matches the given series exactly. For n = 6 the same expression gives 3·6⁴+2·6²−6 = 3888+72−6 = 3954. Hence 3954 is the next term that preserves this quartic-plus-quadratic-minus-linear relationship.

Option A:
Option A, 3924, is 30 less than the value dictated by the formula and does not satisfy aₙ = 3n⁴+2n²−n for n = 6. It would require a sudden downward shift that is not reflected in the behaviour of previous terms. Therefore option A is not consistent with the underlying rule.

Option B:
Option B, 3954, matches exactly the result of applying 3n⁴+2n²−n when n = 6. It continues the same combination of polynomial components that generated each earlier term in the series. Because no change to the algebraic structure is necessary, 3954 is the correct continuation.

Option C:
Option C, 3984, overshoots the correct value by 30 and again does not come from the formula for the sixth term. Choosing 3984 would arbitrarily increase the term and disturb the precise mapping between n and aₙ. Thus option C is not logically acceptable.

Option D:
Option D, 3996, deviates even more from the correct result and has no basis in the expression 3n⁴+2n²−n. Using 3996 would destroy the internal consistency of the series and cannot be considered correct.