In the number series 2, 10, 30, 68, 130, the next term is _______. – UGC NET Question

The nth term of this series is given by the expression (a_n = n^3 + n) with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 1³+1 = 2, 2³+2 = 10, 3³+3 = 30, 4³+4 = 68 and 5³+5 = 130, matching the given terms. For n = 6, the expression yields 6³+6 = 216+6 = 222. Hence 222 is the term that keeps the sequence consistent with a single cubic formula.

Option A:
Option A, 214, does not arise from (n^3+n) for any integer n that logically follows 1 to 5 in this context. Choosing 214 would force a change in the algebraic rule that explains all earlier terms. Therefore 214 is not compatible with the series.

Option B:
Option B, 218, is slightly below the value given by 6³+6 and cannot be expressed as (n^3+n) with an integer n continuing the order. This inconsistency shows that 218 does not respect the functional pattern behind the sequence.

Option C:
Option C, 220, lies between 218 and 222 but still does not equal 6³+6. Using 220 would abandon the clean polynomial structure and introduce an arbitrary number. Hence 220 cannot be treated as the correct continuation.

Option D:
Option D, 222, matches exactly the value produced by substituting n = 6 into (a_n = n^3+n). Because this single expression generates all the given terms and 222 as well, this option preserves the mathematical relationship perfectly and is therefore correct.