In the number series 4, 49, 298, 1153, 3376, the next term is _______. – UGC NET Question

This sequence is generated by the rule aₙ = n⁵ + 2n³ + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 4, 49, 298, 1153 and 3376, matching the given terms exactly. Evaluating the same polynomial for n = 6 gives 8209. Therefore 8209 is the unique next term that preserves this quintic-plus-cubic structure.

Option A:
Option A, 8185, is 24 less than the value produced by aₙ = n⁵ + 2n³ + 1 at n = 6. Adopting 8185 would require lowering the correct polynomial result only at this term. Such an arbitrary correction is not reflected in earlier values, so option A cannot be correct.

Option B:
Option B, 8209, equals exactly the number obtained when n = 6 is substituted into the rule. It maintains the same higher-degree dependence on n that explains the rapid growth of the series. Because no change in coefficients or constants is needed, 8209 correctly continues the number series.

Option C:
Option C, 8225, is slightly greater than the pattern value and does not satisfy the formula for n = 6. Choosing 8225 would add an unjustified extra amount to the correct term, breaking the clean algebraic rule. Thus this option is not a valid continuation.

Option D:
Option D, 8241, deviates even more from the expected value and likewise cannot be produced by n⁵ + 2n³ + 1 at n = 6. Using 8241 would destroy the precise mapping between index and term that characterises the series. Therefore option D is incorrect.