In the number series 6, 60, 252, 720, 1650, the next term is _______. – UGC NET Question

The series is described by the formula aₙ = 2n⁴ + 3n³ + n² with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2+3+1 = 6, 32+24+4 = 60, 162+81+9 = 252, 512+192+16 = 720 and 1250+375+25 = 1650. For n = 6 this expression yields 2·6⁴+3·6³+6² = 2592+648+36 = 3276. Thus 3276 is the unique next term that maintains this quartic-plus-cubic-plus-quadratic structure.

Option A:
Option A, 3210, is smaller than the formula’s result and does not satisfy aₙ = 2n⁴+3n³+n² for n = 6. It would require subtracting 66 from the correctly computed value at the last term only. Because the earlier terms follow the rule exactly, such a modification is unjustified and makes option A incorrect.

Option B:
Option B, 3276, matches precisely the value computed from the expression for n = 6. It preserves all polynomial components and coefficients that describe the growth of the series from the first to the fifth term. Since the same rule generates the sixth term without alteration, 3276 is the correct continuation.

Option C:
Option C, 3330, overshoots the formula’s output and cannot be obtained by substituting n = 6 into the rule. Choosing 3330 would mean arbitrarily inflating the sixth term, which breaks the tight correspondence between index and value. Therefore option C is incorrect.

Option D:
Option D, 3396, deviates even more from the correct value and has no basis in the expression 2n⁴+3n³+n² for the given index. Using 3396 would destroy the internal consistency of the polynomial pattern, so option D is incorrect.