In the number series 4, 46, 236, 754, 1852, the next term is _______. – UGC NET Question

This sequence is defined by the rule (a_n = 3n^4 - n^2 + 2) with n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 3−1+2 = 4, 48−4+2 = 46, 243−9+2 = 236, 768−16+2 = 754 and 1875−25+2 = 1852. For n = 6 we calculate (3×6^4 - 6^2 + 2 = 3888 - 36 + 2 = 3854). Therefore 3854 is the correct next term.

Option A:
Option A, 3854, is exactly the value produced by the formula for n = 6. It preserves the quartic and quadratic contributions in the same way as for the earlier terms and keeps the pattern coherent. Thus 3854 is the correct continuation of the series.

Option B:
Option B, 3818, is 36 less than the correct value and does not equal (3×6^4 - 6^2 + 2). It suggests an unwarranted reduction that is not reflected in the previous behaviour. Hence 3818 is not valid.

Option C:
Option C, 3886, overshoots the computed value and again fails to satisfy the formula. Choosing 3886 would break the algebraic structure and introduce an arbitrary jump. Therefore 3886 is not correct.

Option D:
Option D, 3922, deviates even more from 3854 and has no basis in the generating expression. Using 3922 would destroy the close match between the rule and the data, so it is not the right answer.