In the number series 6, 24, 60, 120, 210, the next term is _______. – UGC NET Question

The nth term of the series can be described by the formula (a_n = n^3 - n) with n starting from 2. Substituting n = 2, 3, 4, 5 and 6 yields 2³−2 = 6, 3³−3 = 24, 4³−4 = 60, 5³−5 = 120 and 6³−6 = 210. For n = 7, the expression gives 7³−7 = 343−7 = 336. Therefore 336 is the unique value that preserves this cubic relationship.

Option A:
Option A, 336, matches exactly the value produced by substituting n = 7 into (a_n = n^3-n). Because the same formula generates all earlier terms and 336 as well, the pattern remains perfectly intact. Hence 336 is the correct continuation of the sequence.

Option B:
Option B, 340, does not equal (7^3-7) and cannot be obtained from the same formula without altering it. Using 340 would break the neat polynomial structure that explains the given terms. Therefore 340 is not consistent with the pattern.

Option C:
Option C, 342, similarly fails to satisfy the expression (n^3-n) for an integer n following the observed order. It would require changing the rule solely for the last term, which is mathematically artificial. Hence 342 cannot be accepted.

Option D:
Option D, 346, lies even further away from the predicted value and has no direct connection with the defining cubic relation. Selecting 346 would abandon the underlying functional rule, so it is not the correct next term.