In the number series 2, 41, 226, 737, 1826, the next term is _______. – UGC NET Question

The terms follow the pattern aₙ = 3n⁴ − 2n² + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 3·1⁴−2·1²+1 = 2, 3·2⁴−2·2²+1 = 41, 3·3⁴−2·3²+1 = 226, 3·4⁴−2·4²+1 = 737 and 3·5⁴−2·5²+1 = 1826, exactly matching the given sequence. For n = 6 this formula gives 3·6⁴−2·6²+1 = 3·1296−72+1 = 3888−72+1 = 3817. Hence 3817 is the unique next term that keeps the polynomial rule intact.

Option A:
Option A, 3817, is precisely the value obtained from 3n⁴−2n²+1 when n = 6. It continues the same quartic pattern that accurately generates all of the previous terms. Because no change in coefficients or constants is needed, this option is fully consistent with the structure of the sequence.

Option B:
Option B, 3805, is smaller than the formula’s output and does not satisfy 3·6⁴−2·6²+1. Choosing 3805 would require subtracting an extra 12 only at the final term. Such an arbitrary adjustment breaks the clean algebraic relationship and therefore cannot be correct.

Option C:
Option C, 3825, is larger than the correct value and again does not arise from substituting n = 6 into the rule. It would imply inflating the term by 8 compared to the polynomial prediction. Since the earlier terms fit the rule exactly, this deviation is unjustified and makes option C wrong.

Option D:
Option D, 3841, deviates even further from the formula’s result. It has no basis in the expression 3n⁴−2n²+1 for the next index and would destroy the precise mapping between n and aₙ. Therefore option D is not a valid continuation of the given series.