In the number series 3, 16, 45, 96, 175, the next term is _______. – UGC NET Question

The series follows the formula aₙ = n³+2n² for n starting from 1. For n = 1, 2, 3, 4 and 5 we calculate 1+2 = 3, 8+8 = 16, 27+18 = 45, 64+32 = 96 and 125+50 = 175. For n = 6 the expression gives 216+72 = 288. Hence 288 is the term that correctly extends this cubic-plus-square pattern.

Option A:
Option A, 278, is 10 less than the required value and does not equal n³+2n² for n = 6. It breaks the exact numeric relationship that has generated all previous terms. Therefore 278 is not the correct continuation.

Option B:
Option B, 282, is closer but still not equal to 216+72. It would imply a small but unjustified alteration of the formula at the last step. Thus 282 does not fit the pattern.

Option C:
Option C, 296, exceeds the computed value and again does not come from n³+2n² when n = 6. Choosing 296 would destroy the perfectly consistent algebraic structure of the sequence. Hence it is not valid.

Option D:
Option D, 288, is exactly 216+72 and comes directly from the rule aₙ = n³+2n². It preserves both the cubic and quadratic contributions in the same way as before, making 288 the correct next term.