In the number series 3, 38, 255, 1044, 3155, the next term is _______. – UGC NET Question

This sequence is described by the rule aₙ = n⁵ + n² + n for n starting from 1. For n = 1, 2, 3, 4 and 5 the expression produces 3, 38, 255, 1044 and 3155, which reproduces the given series exactly. Applying the same rule for n = 6 yields 7818. Therefore 7818 is the unique next term that keeps this quintic pattern intact.

Option A:
Option A, 7766, is smaller than the value from aₙ = n⁵ + n² + n at n = 6. To obtain 7766 we would need to reduce the correct result by 52 only for the sixth term. Such a sudden change is not suggested by earlier terms, so 7766 is inconsistent with the established rule.

Option B:
Option B, 7790, is closer but still not equal to the expression’s value for n = 6. It would require subtracting 28 from the polynomial outcome, which again alters the pattern at only one point. Because the formula matches all previous terms exactly, 7790 cannot correctly continue the series.

Option C:
Option C, 7806, is 12 less than the required value and likewise fails to satisfy aₙ = n⁵ + n² + n when n = 6. Accepting 7806 would mean slightly lowering the correct quintic result at the last step without justification. Hence option C is not compatible with the underlying pattern.

Option D:
Option D, 7818, coincides exactly with the value obtained from the rule for n = 6. It preserves the combination of fifth-power and square growth plus the linear term that explains the earlier numbers. Because it extends the same algebraic structure without any adjustment, 7818 is the correct answer.