In the number series 4, 10, 31, 88, 223, the next term is _______. – UGC NET Question

The series can be modelled by the recurrence a₁ = 4 and aₙ₊₁ = 2aₙ + 3n² − 1 for n ≥ 1. Applying this rule gives a₂ = 2·4+3·1²−1 = 10, a₃ = 2·10+3·2²−1 = 31, a₄ = 2·31+3·3²−1 = 88 and a₅ = 2·88+3·4²−1 = 223, which matches the given sequence. For n = 5 the next term is a₆ = 2·223+3·5²−1 = 446+75−1 = 520. Therefore 520 is the unique next term that obeys this recurrence.

Option A:
Option A, 520, is exactly the value produced by the rule aₙ₊₁ = 2aₙ+3n²−1 when applied to a₅ = 223 with n = 5. It preserves the same structure of doubling the previous term and adding a square-based correction. Since all previous terms are generated in exactly this way, 520 is the correct continuation.

Option B:
Option B, 512, is slightly smaller than the recurrence output and does not satisfy 2·223+3·5²−1. It would require subtracting 8 from the correct value only at the final step. This arbitrary change is not reflected earlier in the sequence, so option B is not consistent with the pattern.

Option C:
Option C, 528, is larger than the correctly computed value and cannot be derived from the recurrence without altering its parameters. Choosing 528 would artificially inflate the sixth term and destroy the exact match between the rule and the data. Hence option C is incorrect.

Option D:
Option D, 536, deviates even more from 520 and again fails to arise from the relation aₙ₊₁ = 2aₙ+3n²−1. Using 536 would break the carefully defined index-dependent correction term, so this option is incorrect.