In the number series 12, 39, 124, 327, 732, the next term is _______. – UGC NET Question

The series follows the rule aₙ = n⁴ + 4n² + 7 with n beginning at 1. Evaluating this expression for n = 1, 2, 3, 4 and 5 gives 12, 39, 124, 327 and 732, which are exactly the given terms. For n = 6 we get a₆ = 6⁴ + 4·6² + 7 = 1296 + 4·36 + 7 = 1296 + 144 + 7 = 1447. Thus 1447 is the unique number that fits this quartic-plus-quadratic pattern.

Option A:
Option A, 1447, is precisely the value produced by n⁴ + 4n² + 7 when n = 6. It maintains the same combination of a dominant fourth-power term and quadratic correction plus constant that explains earlier terms. Since no deviation from the rule is required, this option correctly continues the series.

Option B:
Option B, 1423, is 24 less than the formula’s value at n = 6 and cannot be generated by the same expression. To obtain 1423 we would have to subtract 24 only at the last term, which is inconsistent with the pattern. Therefore option B is not correct.

Option C:
Option C, 1435, is 12 below the required value 1447 and again fails to equal 6⁴ + 4·6² + 7. It offers an approximate but incorrect continuation that ignores the exact algebraic relationship. Hence this option must be rejected.

Option D:
Option D, 1459, is 12 greater than the computed term and would demand increasing the expression’s value only at the sixth position. Such a change would destroy the perfect fit between formula and data, so option D does not represent the correct next term.