In the number series 8, 27, 86, 267, 818, the next term is _______. – UGC NET Question

This sequence is defined recursively by a₁ = 8 and aₙ₊₁ = 3aₙ + 2ⁿ + 1 for n ≥ 1. Using this rule gives a₂ = 3·8 + 2¹ + 1 = 27, a₃ = 3·27 + 2² + 1 = 86, a₄ = 3·86 + 2³ + 1 = 267 and a₅ = 3·267 + 2⁴ + 1 = 818, which exactly matches the given terms. For n = 5 the next term is a₆ = 3·818 + 2⁵ + 1 = 2454 + 32 + 1 = 2487. Hence 2487 is the unique value that satisfies this recurrence and continues the series correctly.

Option A:
Option A, 2463, is 24 less than the value obtained from 3a₅ + 2⁵ + 1 and cannot be generated by the recurrence. To get 2463 we would have to reduce the computed result only at this step, breaking the consistent update rule. Therefore option A is not correct.

Option B:
Option B, 2475, is 12 smaller than the correct term 2487 and again does not equal 3·818 + 32 + 1. It approximates the right value but does not follow from the recurrence formula. As such, option B cannot be considered a valid continuation of the sequence.

Option C:
Option C, 2487, exactly matches the value produced by the recurrence for n = 5. It maintains the structure of tripling the previous term and adding a power of 2 plus 1, which explains all prior transitions. Because it respects the rule fully, this option is the correct answer.

Option D:
Option D, 2499, is 12 greater than the correct value and would require increasing the additive part beyond 2⁵ + 1 at the final step. This change would alter the recurrence mechanism only at a single position and is not supported by earlier terms. Hence option D is incorrect.