In the number series 6, 57, 262, 801, 1926, the next term is _______. – UGC NET Question

This sequence is generated by the polynomial aₙ = 3n⁴ + 2n² + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 this rule gives 6, 57, 262, 801 and 1926, exactly matching the given terms. Using the same expression for n = 6 produces 3961. Thus 3961 is the only value that maintains this polynomial pattern consistently.

Option A:
Option A, 3949, is smaller than the value produced by 3n⁴ + 2n² + 1 for n = 6. To accept 3949 we would have to reduce the correct polynomial result by 12 only at the sixth term. Such a selective adjustment is not supported by the earlier terms, so this option is incorrect.

Option B:
Option B, 3961, is exactly the number obtained when n = 6 is substituted into aₙ = 3n⁴ + 2n² + 1. It keeps the same balance between the fourth-power and square terms that explains the growth of the series. Because it extends the rule without modification, 3961 is the correct next term.

Option C:
Option C, 3973, is 12 greater than the pattern value. Accepting 3973 would require adding an arbitrary constant at the final step while leaving the rule unchanged before, which breaks the algebraic consistency. Hence this option does not correctly continue the sequence.

Option D:
Option D, 3997, deviates even more from the required value. It cannot be produced by 3n⁴ + 2n² + 1 at n = 6 and would destroy the tight functional link between index and term. Therefore 3997 is not a valid continuation of the number series.