In the number series 6, 63, 318, 1011, 2478, the next term is _______. – UGC NET Question

The terms are generated by aₙ = 4n⁴ − n² + 3 with n starting at 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 63, 318, 1011 and 2478, which matches the series. For n = 6 we obtain a₆ = 4·6⁴ − 6² + 3 = 4·1296 − 36 + 3 = 5184 − 33 = 5151. Thus 5151 is the only term consistent with this quartic pattern.

Option A:
Option A, 5127, is 24 less than the value predicted by 4n⁴ − n² + 3 at n = 6. Adopting 5127 would mean arbitrarily reducing the correct polynomial result at the final step. Since earlier values fit the rule exactly, this modification is unjustified, so option A is incorrect.

Option B:
Option B, 5139, is 12 less than the correct value 5151 and cannot be obtained from 4·6⁴ − 6² + 3. It suggests a slight downward deviation that does not follow from the formula. Therefore this option fails to provide a mathematically sound continuation of the series.

Option C:
Option C, 5151, is precisely the number produced when n = 6 is substituted into 4n⁴ − n² + 3. It maintains the same balance between the quartic term and the subtractive quadratic component that governs the earlier terms. Because it extends the pattern without any alteration, this option is correct.

Option D:
Option D, 5163, is 12 greater than the polynomial value and likewise cannot be generated by the rule at n = 6. Introducing such an increase at only one term would break the consistent algebraic structure. Hence option D is not a valid continuation.