In the number series 1, 31, 241, 1021, 3121, the next term is _______. – UGC NET Question

This series is generated by the quintic rule aₙ = n⁵ − n + 1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1−1+1 = 1, 32−2+1 = 31, 243−3+1 = 241, 1024−4+1 = 1021 and 3125−5+1 = 3121, exactly matching the given numbers. For n = 6 the same expression gives 6⁵−6+1 = 7776−6+1 = 7771. Therefore 7771 is the unique next term that maintains this fifth-degree pattern.

Option A:
Option A, 7729, is significantly below the computed value and does not satisfy aₙ = n⁵−n+1 for n = 6. It would require subtracting 42 from the formula’s result only at the last term. Such a deviation has no basis in the earlier behaviour of the sequence, so option A is incorrect.

Option B:
Option B, 7759, is closer but still does not equal 7771 and therefore fails to arise from applying n⁵−n+1 to n = 6. Selecting 7759 would alter the functional rule that perfectly explains all previous terms. Hence option B is not a valid continuation of the pattern.

Option C:
Option C, 7771, coincides exactly with 6⁵−6+1 and follows directly from the quintic expression. It keeps the same high-order dependence on n that produces the rapid growth observed in the sequence. Because it continues the rule without modification, 7771 is the correct next term.

Option D:
Option D, 7789, overshoots the formula’s output by 18 and cannot be expressed as n⁵−n+1 for the sixth position. Using 7789 would break the precise relationship between index and term, making option D inconsistent with the established series.