In the number series 4, 37, 130, 313, 616, the next term is _______. – UGC NET Question

This sequence is described by the rule (a_n = 5n^3 - 2n + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 4, 37, 130, 313 and 616, matching the given terms. For n = 6 we compute (5×6^3 - 2×6 + 1 = 1080 - 12 + 1 = 1069). Therefore 1069 is the correct next term.

Option A:
Option A, 1047, is smaller than the formula’s result and does not equal (5×6^3 - 2×6 + 1). It suggests a reduced value for no reason indicated by the pattern. Hence 1047 is not a valid continuation.

Option B:
Option B, 1059, is closer but still fails to match the computed value of 1069. Choosing 1059 would alter the algebraic rule only at the last step, which is inconsistent. Thus 1059 is not correct.

Option C:
Option C, 1069, coincides exactly with the output of the rule for n = 6. It preserves the combination of cubic and linear terms that generate all earlier elements. For this reason, 1069 is the correct continuation.

Option D:
Option D, 1087, overshoots the formula’s value and cannot be obtained from (5n^3 - 2n + 1) when n = 6. Adopting 1087 would destroy the precise structure of the series, so it is not appropriate.