In the number series 4, 65, 486, 2047, 6248, the next term is _______. – UGC NET Question

The series follows the rule aₙ = 2n⁵ − n + 3 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 4, 65, 486, 2047 and 6248, which are exactly the terms in the question. Using the same formula for n = 6 produces 15549. Hence 15549 is the only value that continues this quintic pattern without altering the rule.

Option A:
Option A, 15549, is precisely the value obtained from aₙ = 2n⁵ − n + 3 when n = 6. It maintains the strong fifth-power growth together with the linear correction seen in earlier terms. Because it emerges directly from the established expression, this option correctly represents the next term.

Option B:
Option B, 15521, is 28 less than the formula’s output at n = 6. Accepting 15521 would require arbitrarily lowering the quintic value only for this term. Since all previous entries fit the rule exactly, such an adjustment is unjustified and makes option B incorrect.

Option C:
Option C, 15581, is slightly larger than the required value and does not equal 2n⁵ − n + 3 when n = 6. It would introduce a positive discrepancy at the last step that is not present earlier. Therefore this option breaks the algebraic consistency of the sequence.

Option D:
Option D, 15613, deviates even more from the computed value and cannot be produced by the same formula. Using 15613 would destroy the exact match between the rule and the observed terms, so it is not a valid continuation.