In the number series 6, 48, 238, 756, 1854, the next term is _______. – UGC NET Question

The pattern here is given by aₙ = 3n⁴ − n² + 4 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 yields 6, 48, 238, 756 and 1854, which are exactly the numbers in the series. Applying the same expression for n = 6 gives 3856. Hence 3856 is the only value that keeps this quartic-based pattern consistent.

Option A:
Option A, 3824, is 32 less than the value obtained from aₙ = 3n⁴ − n² + 4 when n = 6. To accept 3824 we would need to subtract 32 only at the last term, which is not indicated by the earlier entries. As a result, this option does not preserve the polynomial rule and is incorrect.

Option B:
Option B, 3840, is still smaller than the required value and does not equal the output of the formula at n = 6. It represents a partial correction toward the right term but still breaks the functional relationship. Therefore option B cannot correctly continue the series.

Option C:
Option C, 3856, matches exactly the value predicted by 3n⁴ − n² + 4 for n = 6. It respects the same combination of a fourth-power term and a negative square adjustment that shapes the growth of the sequence. Because the same rule explains all the given terms and leads naturally to 3856, this option is correct.

Option D:
Option D, 3872, is 16 greater than the polynomial value and cannot be generated by the rule for the sixth position. Introducing this extra amount would break the algebraic pattern that the series follows. Thus option D is not a valid continuation.