In the number series 5, 12, 31, 68, 129, the next term is _______. – UGC NET Question

This series follows the rule (a_n = n^3 + 4) for n starting from 1. Substituting n=1,2,3,4 and 5 gives 1³+4 = 5, 2³+4 = 12, 3³+4 = 31, 4³+4 = 68 and 5³+4 = 129, which matches the given terms. For the next term, n=6, so we calculate 6³+4 = 216+4 = 220. Hence 220 is the correct next term.

Option A:
Option A, 214, does not correspond to (n^3+4) for any integer n that continues naturally after the existing values. Choosing 214 would force a change in the underlying formula. Therefore it is not compatible with the pattern.

Option B:
Option B, 218, is slightly smaller than the value produced by 6³+4 and again cannot be expressed using the same rule. This discrepancy shows that 218 does not fit the algebraic structure of the series.

Option C:
Option C equals 220, which exactly matches the value of (n^3+4) when n=6. It preserves the single formula that generates all earlier terms and therefore extends the sequence in a logically consistent manner.

Option D:
Option D, 226, is larger than the cubic prediction and does not satisfy the relation (a_n = n^3+4) for any appropriate integer n in order. As a result, 226 cannot be considered the correct continuation.