In the number series 2, 12, 36, 80, 150, the next term is _______. – UGC NET Question

This sequence is generated by the formula (a_n = n^3 + n^2) with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1³+1² = 2, 2³+2² = 12, 3³+3² = 36, 4³+4² = 80 and 5³+5² = 150. For n = 6 the formula gives 6³+6² = 216+36 = 252. Thus 252 maintains a single algebraic rule that explains all the terms.

Option A:
Option A, 252, matches exactly the value obtained by substituting n = 6 into the rule (a_n = n^3+n^2). Since this rule already accounts for every earlier term, extending it to 252 keeps the series perfectly consistent. Hence 252 is the correct continuation.

Option B:
Option B, 254, cannot be expressed as 6³+6² and does not result from the same expression for any natural n in order. Using 254 would break the clean connection between term position and value. Therefore it is not the right answer.

Option C:
Option C, 256, is numerically close but again does not obey the formula (n^3+n^2). It has no direct link to the pattern that generates the given sequence and so is not acceptable.

Option D:
Option D, 258, deviates even further from the predicted value and similarly lacks a basis in the generating rule. Selecting 258 would force an ad hoc change in the pattern, making it incorrect as the next term.