In the number series 5, 60, 333, 1232, 3525, the next term is _______. – UGC NET Question

This sequence is described by the rule aₙ = n⁵ + 3n³ + n² with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 5, 60, 333, 1232 and 3525, exactly matching the given numbers. For n = 6, we compute a₆ = 6⁵ + 3·6³ + 6² = 7776 + 3·216 + 36 = 7776 + 648 + 36 = 8460. Therefore 8460 is the only value that keeps this polynomial pattern intact.

Option A:
Option A, 8436, is 24 less than the value predicted by n⁵ + 3n³ + n² at n = 6. To accept 8436 we would have to subtract 24 from the correct result only at this term, despite perfect agreement elsewhere. This change would break the algebraic rule, so option A is incorrect.

Option B:
Option B, 8460, exactly equals the output of the formula for n = 6. It continues the same combination of a dominant fifth-power term with cubic and quadratic corrections that shape the earlier terms. Because it arises directly from the established expression, this option correctly extends the series.

Option C:
Option C, 8448, is 12 less than the correct value and cannot be written as 6⁵ + 3·6³ + 6². It reflects a small downward shift that is not supported by any structural change in the rule. Hence this option does not provide a valid continuation of the pattern.

Option D:
Option D, 8472, is 12 greater than the required value and would similarly require inflating the polynomial result at the sixth term. This violates the consistency of the relationship between n and aₙ, so option D must be rejected.