In the number series 5, 38, 173, 530, 1277, the next term is _______. – UGC NET Question

The pattern here is aₙ = 2n⁴ + n² + 2 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 2·1⁴+1²+2 = 5, 2·2⁴+2²+2 = 38, 2·3⁴+3²+2 = 173, 2·4⁴+4²+2 = 530 and 2·5⁴+5²+2 = 1277, which confirms the rule. For n = 6 we have 2·6⁴+6²+2 = 2·1296+36+2 = 2592+38 = 2630. Hence 2630 is the only value that continues this quartic-plus-quadratic-plus-constant pattern.

Option A:
Option A, 2598, is 32 less than the formula’s outcome and does not satisfy aₙ = 2n⁴+n²+2 for n = 6. It would introduce an arbitrary drop at the sixth term that is not suggested by the earlier values. Thus option A is not consistent with the sequence.

Option B:
Option B, 2614, is closer but still fails to match the value 2630 predicted by the rule. Choosing 2614 would mean altering the pattern only at the final step, which is logically unjustified. Therefore option B cannot be correct.

Option C:
Option C, 2622, also differs from the formula’s result and cannot be obtained from substituting n = 6 into 2n⁴+n²+2. It represents a partial correction but still does not maintain the exact algebraic structure. As a result, option C is not a valid continuation.

Option D:
Option D, 2630, matches exactly the computed value for n = 6. It preserves the dependence on n⁴ and n², as well as the constant term, in exactly the same way as the earlier terms. Because the same rule explains every entry including the next, 2630 is the correct answer.