In the number series 6, 29, 82, 177, 326, the next term is _______. – UGC NET Question

This series is given by the rule (a_n = 2n^3 + 3n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 6, 29, 82, 177 and 326, which match the listed terms. For n = 6 we compute (2×6^3 + 3×6^2 + 1 = 432 + 108 + 1 = 541). Thus 541 is the unique next term that maintains this cubic-plus-square pattern.

Option A:
Option A, 529, is 12 less than the formula’s result and does not equal (2×6^3 + 3×6^2 + 1). It would imply altering the rule at the last term, which is not supported by the sequence. Therefore 529 is not correct.

Option B:
Option B, 541, coincides exactly with the value produced by the expression for n = 6. It preserves all the coefficients and the constant term used for earlier values. For this reason, 541 is the correct continuation.

Option C:
Option C, 547, overshoots the computed value and cannot be written as (2n^3 + 3n^2 + 1) for the next index. Choosing 547 would break the algebraic consistency, so it is not valid.

Option D:
Option D, 553, deviates even more from the formula and again fails to satisfy the same relationship for n = 6. Adopting 553 would distort the pattern, so it is not the right answer.