In the number series 6, 45, 190, 561, 1326, the next term is _______. – UGC NET Question

The series can be generated by the formula (a_n = 2n^4 + 3n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 6, 45, 190, 561 and 1326, which match the question. For n = 6 the expression gives (2×6^4 + 3×6^2 + 1 = 2592 + 108 + 1 = 2701). Hence 2701 is the correct next term.

Option A:
Option A, 2665, is 36 less than the formula’s output and does not equal (2×6^4 + 3×6^2 + 1). It would require an unjustified reduction at the final term, so 2665 is not valid.

Option B:
Option B, 2683, remains below 2701 and fails to satisfy the expression for n = 6. Choosing 2683 would disrupt the quartic-plus-square structure of the sequence. Thus 2683 is not correct.

Option C:
Option C, 2723, overshoots the computed value and again cannot be obtained from the given formula. Using 2723 would break the algebraic pattern that has held so far. Hence 2723 is not appropriate.

Option D:
Option D, 2701, exactly equals the value generated by the rule for n = 6. It maintains the same functional relationship between n and a_n for all terms, so 2701 is the correct continuation of the series.