In the number series 5, 31, 97, 221, 421, the next term is _______. – UGC NET Question

The terms satisfy the rule (a_n = 3n^3 + 2n^2 - n + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 this expression produces 5, 31, 97, 221 and 421, matching the series. For n = 6 we compute (3×6^3 + 2×6^2 - 6 + 1 = 648 + 72 - 6 + 1 = 715). Hence 715 is the correct next term.

Option A:
Option A, 703, is 12 less than the value given by the formula and does not equal (3×6^3 + 2×6^2 - 6 + 1). It implies a pattern change that is not supported by earlier terms. Therefore 703 is not valid.

Option B:
Option B, 709, is still below 715 and cannot be obtained from the same expression for n = 6. Selecting 709 would break the algebraic rule that explains the sequence. Thus 709 is not correct.

Option C:
Option C, 721, overshoots the computed value and likewise fails to match the formula. It suggests an arbitrary increase not grounded in the pattern. Hence 721 is not appropriate.

Option D:
Option D, 715, equals the exact output of the rule for n = 6 and preserves both the cubic and quadratic contributions of the expression. For this reason, 715 is the correct continuation of the series.