In the number series 3, 21, 73, 177, 351, the next term is _______. – UGC NET Question

The terms follow the rule (a_n = 3n^3 - n^2 + 1) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 21, 73, 177 and 351, which match the series exactly. For n = 6 we compute (3×6^3 - 6^2 + 1 = 648 - 36 + 1 = 613). Therefore 613 is the value that continues the same cubic pattern.

Option A:
Option A, 601, is smaller than the value obtained from the formula and does not equal (3×6^3 - 6^2 + 1). It would imply subtracting an extra 12 without any justification in the rule. Hence 601 is not consistent with the sequence.

Option B:
Option B, 607, is still below 613 and again cannot be derived from the expression for n = 6. Choosing 607 would break the precise connection between index and term. Thus 607 is not a valid continuation.

Option C:
Option C, 613, exactly equals the result of applying (a_n = 3n^3 - n^2 + 1) with n = 6. It preserves both the cubic and quadratic contributions that explain all previous terms. For this reason, 613 is the correct next term.

Option D:
Option D, 625, overshoots the formula’s prediction and fails to satisfy the same expression for n = 6. Adopting 625 would change the rule at the last step and disrupt the pattern, so it is not correct.