In the number series 10, 49, 194, 565, 1330, the next term is _______. – UGC NET Question

This series follows the rule aₙ = 2n⁴ + 3n² + 5 with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 10, 49, 194, 565 and 1330, exactly reproducing the given terms. When n = 6, we obtain a₆ = 2·6⁴ + 3·6² + 5 = 2·1296 + 3·36 + 5 = 2592 + 108 + 5 = 2705. Thus 2705 is the only value that fits this polynomial rule and correctly continues the series.

Option A:
Option A, 2681, is 24 less than the value produced by 2n⁴ + 3n² + 5 at n = 6. Taking 2681 would require lowering the correct outcome only at the sixth term, which is inconsistent with the exact agreement at earlier positions. Hence this option does not preserve the algebraic pattern.

Option B:
Option B, 2705, exactly matches the result of substituting n = 6 into the rule 2n⁴ + 3n² + 5. It respects both the fourth-power component and the quadratic correction that govern the growth of the sequence. Because it extends the same functional relationship without alteration, this option is the correct answer.

Option C:
Option C, 2693, is slightly below the correct value and cannot be expressed as 2·6⁴ + 3·6² + 5. Selecting 2693 would amount to undercounting the contribution of the polynomial terms at the final step. Therefore it does not provide a legitimate continuation of the series.

Option D:
Option D, 2717, is 12 greater than the required value and would similarly require modifying the rule only for n = 6. Such a change is not justified by the previous terms, where the formula holds exactly. Consequently, option D is not a valid extension of the given pattern.