In the number series 3, 12, 33, 72, 135, the next term is _______. – UGC NET Question

This sequence follows the rule aₙ = n³+2n for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1³+2 = 3, 2³+4 = 12, 3³+6 = 33, 4³+8 = 72 and 5³+10 = 135, matching the given terms. For n = 6 the formula gives 6³+12 = 216+12 = 228. Therefore 228 is the only value that continues the same cubic relationship without any adjustment.

Option A:
Option A, 218, does not equal n³+2n for n = 6 and would require subtracting 10 from the value given by the formula. This distortion breaks the clear pattern connecting each term to its index. Thus 218 cannot be regarded as the correct next term.

Option B:
Option B, 222, is still below the correct cubic value of 228 and does not satisfy the rule aₙ = n³+2n. It represents an arbitrary modification rather than a logical extension of the pattern. For this reason it is not the right answer.

Option C:
Option C, 228, is precisely the value obtained from the generating rule when n = 6. It respects the same algebraic structure used for all earlier terms and keeps the series internally consistent. Hence 228 is the correct continuation.

Option D:
Option D, 236, exceeds the formula’s prediction and has no basis in the expression n³+2n for the next index. Adopting 236 would undermine the transparent cubic pattern, so it is not a valid option.