In the number series 6, 23, 90, 267, 638, the next term is _______. – UGC NET Question

The sequence follows the formula (a_n = n^4 + 2n + 3) with n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1+2+3 = 6, 16+4+3 = 23, 81+6+3 = 90, 256+8+3 = 267 and 625+10+3 = 638. For n = 6 the expression yields (6^4 + 12 + 3 = 1296 + 15 = 1311). Therefore 1311 is the correct next term under this quartic-plus-linear pattern.

Option A:
Option A, 1287, is 24 less than the calculated value and does not equal (6^4 + 2×6 + 3). It would require weakening the rule only at the final term, which is inconsistent. Hence 1287 is not valid.

Option B:
Option B, 1299, is still below 1311 and again cannot be written as (n^4 + 2n + 3) for n = 6. Adopting 1299 would disrupt the tight connection between index and term. Thus 1299 is not correct.

Option C:
Option C, 1311, matches exactly the output of the formula when n = 6. It preserves the same quartic and linear structure that generated all earlier values. For this reason, 1311 is the correct continuation.

Option D:
Option D, 1327, overshoots the correct value and fails to satisfy the expression for n = 6. Using 1327 would abandon the algebraic coherence of the series, so it is not a valid answer.