In the number series 3, 16, 84, 429, 2161, the next term is _______. – UGC NET Question

This sequence is defined by a₁ = 3 and aₙ₊₁ = 5aₙ + n² for n ≥ 1. Using the rule, we get a₂ = 5·3 + 1² = 16, a₃ = 5·16 + 2² = 84, a₄ = 5·84 + 3² = 429 and a₅ = 5·429 + 4² = 2161, which matches the given terms. For n = 5 the next term is a₆ = 5·2161 + 5² = 10805 + 25 = 10830. Hence 10830 is the only number consistent with this recurrence relation.

Option A:
Option A, 10830, is precisely the value obtained from 5a₅ + 5² when a₅ = 2161. It maintains the pattern of multiplying the previous term by five and adding the square of the index. Since this mechanism generates all earlier terms correctly, 10830 is the valid continuation of the series.

Option B:
Option B, 10806, is 24 less than the computed value and cannot be produced by the recurrence at n = 5. Accepting 10806 would require subtracting 24 from the result only at this stage, which has no basis in the rule. Therefore option B is incorrect.

Option C:
Option C, 10818, is 12 less than the correct value and again fails to equal 5·2161 + 25. It is merely an approximate candidate that does not arise from the stated recurrence. As a result, option C cannot be the correct next term.

Option D:
Option D, 10842, is 12 greater than the required result and would demand increasing the formula output only for the sixth term. This contradicts the consistent pattern established by previous steps, so option D is not a valid continuation.