In the number series 9, 42, 177, 534, 1281, the next term is _______. – UGC NET Question

This series is governed by the rule aₙ = 2n⁴ + n² + 6 with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 9, 42, 177, 534 and 1281, which perfectly matches the given sequence. For n = 6, the formula yields a₆ = 2·6⁴ + 6² + 6 = 2·1296 + 36 + 6 = 2592 + 42 = 2634. Therefore 2634 is the only value that continues this quartic-based pattern correctly.

Option A:
Option A, 2610, is 24 less than the value given by 2n⁴ + n² + 6 at n = 6. To choose 2610 we would have to reduce the correct result at this stage, despite the formula fitting earlier terms exactly. This inconsistency makes option A incorrect.

Option B:
Option B, 2622, is 12 less than the correct answer and likewise does not equal 2·6⁴ + 6² + 6. It provides a close but still inaccurate figure that breaks the exact functional relation. Hence option B cannot be accepted as the next term.

Option C:
Option C, 2646, is 12 greater than the correct value 2634 and cannot be obtained from the rule at n = 6. Accepting 2646 would amount to inflating the polynomial output at only one position, which is not supported by the data. Thus option C is not a valid continuation of the number series.

Option D:
Option D, 2634, matches exactly the value produced by the formula for n = 6. It keeps the same contributions from the quartic term, the square term and the constant used earlier. Because this number arises directly from the established pattern, option D is the correct answer.