In the number series 2, 12, 75, 458, 2763, the next term is _______. – UGC NET Question

This sequence uses the recurrence a₁ = 2 and aₙ₊₁ = 6aₙ + n² − 1 for n ≥ 1. With this rule we obtain a₂ = 6·2 + 1² − 1 = 12, a₃ = 6·12 + 2² − 1 = 75, a₄ = 6·75 + 3² − 1 = 458 and a₅ = 6·458 + 4² − 1 = 2763, perfectly matching the given terms. For n = 5 the next term is a₆ = 6·2763 + 5² − 1 = 16602. Therefore 16602 is the unique next term that satisfies the same recurrence.

Option A:
Option A, 16542, is 60 less than the value produced by aₙ₊₁ = 6aₙ + n² − 1 at n = 5. Selecting 16542 would require subtracting 60 from the properly computed term only at this position. Since earlier transitions obey the rule exactly, this option is inconsistent with the pattern and is incorrect.

Option B:
Option B, 16602, matches exactly the result of applying the recurrence to a₅ with n = 5. It preserves the structure of multiplying the previous term by six and then adding a quadratic correction minus one. Because this mechanism generates all earlier terms and leads straight to 16602, this option correctly continues the series.

Option C:
Option C, 16638, is 36 greater than the correct value and cannot be written as 6·2763 + 5² − 1. To obtain 16638 we would have to increase the additive part beyond n² − 1 at the sixth step, altering the rule. Thus option C does not follow the established recurrence.

Option D:
Option D, 16674, deviates even more from the computed value and similarly fails to satisfy the formula. Using 16674 would destroy the precise relationship between terms that characterises this sequence. Therefore option D is not a valid continuation.