In the number series 6, 21, 69, 218, 674, the next term is _______. – UGC NET Question

The pattern can be expressed recursively with a₁ = 6 and aₙ₊₁ = 3aₙ + 2ⁿ + n for n ≥ 1. Using this rule we get a₂ = 3·6 + 2¹ + 1 = 21, a₃ = 3·21 + 2² + 2 = 69, a₄ = 3·69 + 2³ + 3 = 218 and a₅ = 3·218 + 2⁴ + 4 = 674, which matches the given series. For n = 5 the next term is a₆ = 3·674 + 2⁵ + 5 = 2059. Thus 2059 is the only value consistent with this recurrence.

Option A:
Option A, 2059, is exactly the value obtained from aₙ₊₁ = 3aₙ + 2ⁿ + n when applied to a₅ with n = 5. It keeps the structure of tripling the previous term, adding a power of 2 and then adding the index itself. Because this mechanism reproduces all previous terms and yields 2059 next, this option is correct.

Option B:
Option B, 2035, is 24 less than the recurrence result and cannot be written as 3·674 + 2⁵ + 5. Accepting 2035 would require subtracting 24 from the properly computed term, with no justification from the earlier steps. Therefore this option does not respect the established pattern and is incorrect.

Option C:
Option C, 2073, is 14 greater than the required value and also fails to match the recurrence. To reach 2073 the additive part after 3a₅ would have to be larger than 2⁵ + 5, changing the rule at the last step. Consequently, option C is not a valid continuation.

Option D:
Option D, 2087, deviates even more from 2059 and likewise cannot be produced by the rule. Using 2087 would break the exact connection between consecutive terms that defines the sequence. Hence option D cannot be the correct answer.