In the number series 3, 21, 91, 273, 651, the next term is _______. – UGC NET Question

This sequence follows the rule (a_n = n^4 + n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+1+1 = 3, 16+4+1 = 21, 81+9+1 = 91, 256+16+1 = 273 and 625+25+1 = 651. For n = 6 the formula gives (6^4 + 6^2 + 1 = 1296 + 36 + 1 = 1333). Therefore 1333 is the unique next term that maintains this quartic-plus-square pattern.

Option A:
Option A, 1317, is 16 less than the formula’s result and does not equal (6^4 + 6^2 + 1). It would introduce an unexplained shortfall at the last term. Hence 1317 is not consistent with the series.

Option B:
Option B, 1325, is still lower than 1333 and similarly fails to arise from the expression for n = 6. Choosing 1325 would alter the rule only for the final term, which is logically unsound. Thus 1325 is not a valid continuation.

Option C:
Option C, 1341, overshoots the correct value and cannot be written as (6^4 + 6^2 + 1). It breaks the precise algebraic fit that exists between the terms and their indices. Therefore 1341 is not correct.

Option D:
Option D, 1333, matches exactly the output of the rule (a_n = n^4 + n^2 + 1) when n = 6. It preserves the same quartic and quadratic contributions that generate all earlier terms, so 1333 is the correct next term.