In the number series 6, 65, 298, 897, 2126, the next term is _______. – UGC NET Question

This sequence is generated by the rule aₙ = 3n⁴ + 2n³ + 1 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 into the expression gives 6, 65, 298, 897 and 2126, matching the given terms. For n = 6 we compute a₆ = 3·6⁴ + 2·6³ + 1 = 3·1296 + 2·216 + 1 = 3888 + 432 + 1 = 4321. Therefore 4321 is the only value consistent with this polynomial pattern.

Option A:
Option A, 4297, is 24 less than the formula’s output for n = 6 and cannot be produced by 3n⁴ + 2n³ + 1. Accepting 4297 would require lowering the correct value only at the last step, even though earlier terms follow the rule exactly. Hence option A is not a valid continuation.

Option B:
Option B, 4321, matches exactly the value obtained when n = 6 is substituted into the rule. It preserves the dominance of the quartic term with an added cubic component and a constant, just as in previous terms. Because it extends the sequence in a way that fully respects the algebraic structure, this option is correct.

Option C:
Option C, 4309, is 12 less than the correct term 4321 and again does not equal 3·6⁴ + 2·6³ + 1. It is numerically close but fails to reflect the exact functional relationship, so it cannot be accepted.

Option D:
Option D, 4333, is 12 greater than the proper value and cannot be generated by the same expression at n = 6. Choosing 4333 would create an artificial jump in the series that is not implied by the pattern. Therefore this option is incorrect.