In the number series 2, 18, 87, 280, 745, the next term is _______. – UGC NET Question

This series is generated by the rule aₙ = n⁴ + n! with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1⁴ + 1! = 2, 2⁴ + 2! = 18, 3⁴ + 3! = 87, 4⁴ + 4! = 280 and 5⁴ + 5! = 745, which matches the pattern exactly. For n = 6, a₆ = 6⁴ + 6! = 1296 + 720 = 2016. Therefore 2016 is the only value that continues this factorial-quartic combination correctly.

Option A:
Option A, 1992, is 24 less than the value produced by n⁴ + n! at n = 6. Selecting 1992 would require lowering the term below what the formula prescribes only at this index. Such a change contradicts the exact fit observed in earlier terms, so option A is not correct.

Option B:
Option B, 2004, is 12 smaller than the correct value 2016 and again cannot be written as 1296 + 720. It approximates but does not equal the output of the expression for n = 6. Consequently, option B is not a legitimate continuation of the pattern.

Option C:
Option C, 2016, exactly matches the result obtained from the rule for n = 6. It maintains the same contribution from both the quartic term and the factorial term that gave rise to the earlier numbers. Because it extends this combined pattern consistently, option C is the correct answer.

Option D:
Option D, 2028, is 12 greater than the computed value and cannot arise from the expression n⁴ + n! at n = 6. Adopting 2028 would artificially inflate the last term, disrupting the precise relationship between n and aₙ. Hence option D is not a valid continuation.