In the number series 2, 16, 80, 254, 622, the next term is _______. – UGC NET Question

The sequence is described by the formula aₙ = n⁴−n+2 for n starting from 1. For n = 1, 2, 3, 4 and 5, we get 1−1+2 = 2, 16−2+2 = 16, 81−3+2 = 80, 256−4+2 = 254 and 625−5+2 = 622. For n = 6 the formula gives 1296−6+2 = 1292. Thus 1292 is the correct next term in line with this quartic-minus-linear rule.

Option A:
Option A, 1252, is 40 less than the computed value and does not equal n⁴−n+2 for n = 6. It breaks the exact pattern that has held for the earlier terms. Therefore 1252 is not a valid continuation.

Option B:
Option B, 1268, is still lower than 1292 and likewise fails to come from the generating expression. Selecting 1268 would require changing the formula without any clear reason. Hence it is not correct.

Option C:
Option C, 1292, matches perfectly with 6⁴−6+2 and continues the same structural relationship between n and aₙ. It maintains both the power and linear components consistently, making 1292 the correct answer.

Option D:
Option D, 1322, overshoots the correct value and cannot be obtained by substituting n = 6 into the rule. Using 1322 would distort the pattern and undermine the algebraic coherence of the sequence. Thus it is not appropriate.