In the number series 3, 28, 229, 996, 3079, the next term is _______. – UGC NET Question

The pattern is captured by the polynomial aₙ = n⁵ − 2n² + 4 with n starting from 1. For n = 1, 2, 3, 4 and 5 this formula yields 3, 28, 229, 996 and 3079, exactly the given sequence. Evaluating the same expression at n = 6 gives 7708. Thus 7708 is the only value that continues this quintic pattern consistently.

Option A:
Option A, 7708, is precisely the value obtained from aₙ = n⁵ − 2n² + 4 when n = 6. It maintains the strong fifth-power growth combined with the quadratic subtraction and constant term that shape the earlier numbers. Because this rule explains all five terms and leads directly to 7708 next, this option is correct.

Option B:
Option B, 7684, is 24 less than the required value and does not equal the formula’s output for n = 6. To accept 7684 we would need to diminish the polynomial result only at this index. Such a modification is not supported by the rest of the sequence, so option B is incorrect.

Option C:
Option C, 7724, is 16 greater than the pattern value and again cannot be produced by n⁵ − 2n² + 4 at n = 6. Using 7724 would inflate the term beyond the rule’s prediction, breaking the algebraic structure. Therefore option C is not a valid continuation.

Option D:
Option D, 7740, deviates even more from 7708 and likewise fails to satisfy the expression at n = 6. Adopting 7740 would ignore the exact polynomial relationship linking the terms. Hence option D is not acceptable.