In the number series 2, 17, 82, 257, 626, the next term is _______. – UGC NET Question

Each term of the series can be written as n⁴+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1⁴+1 = 2, 2⁴+1 = 17, 3⁴+1 = 82, 4⁴+1 = 257 and 5⁴+1 = 626. The next term corresponds to n = 6, which gives 6⁴+1 = 1296+1 = 1297. Therefore 1297 is the only value that continues the same quartic pattern.

Option A:
Option A matches the value obtained from the rule aₙ = n⁴+1 when n = 6. It extends the exact same relationship that holds for all the earlier terms of the sequence. Because no change in the functional rule is needed to reach 1297, this option is fully consistent with the observed pattern.

Option B:
Option B, 1257, is close in magnitude but does not equal 6⁴+1 and therefore fails to satisfy the quartic formula for n = 6. Choosing it would require altering the rule that clearly generates the previous terms. Hence it cannot represent the correct continuation of the series.

Option C:
Option C, 1273, similarly does not arise from the expression n⁴+1 for any integer n that follows the existing index order. It breaks the tight link between term position and value that is evident in the given numbers. Thus 1273 is not a valid next term.

Option D:
Option D, 1327, overshoots the correct quartic value and cannot be expressed as 6⁴+1. Adopting 1327 would mean abandoning the neat polynomial structure and introducing an arbitrary adjustment. Therefore it is not the correct answer.