In the number series 6, 36, 126, 336, 750, the next term is _______. – UGC NET Question

This series is described by the rule aₙ = n⁴ + 5n² for n starting from 1. Evaluating for n = 1, 2, 3, 4 and 5 yields 1+5 = 6, 16+20 = 36, 81+45 = 126, 256+80 = 336 and 625+125 = 750. For n = 6 we obtain 6⁴+5·6² = 1296+180 = 1476. Therefore 1476 is the only term that keeps this quartic-plus-quadratic pattern intact for the next position.

Option A:
Option A, 1440, is lower than the formula’s output and does not equal n⁴+5n² for n = 6. It would require subtracting 36 from the correctly computed term only at the sixth step, which is not implied by the earlier values. This inconsistency makes option A incorrect.

Option B:
Option B, 1452, still fails to match 1296+180 and thus does not follow the same expression. Although numerically close, it cannot be produced by the rule for the sixth term. Hence option B would break the algebraic structure of the series and is not acceptable.

Option C:
Option C, 1464, lies between some candidate values but again does not satisfy aₙ = n⁴+5n² when n = 6. Selecting 1464 would mean slightly reducing the correct result without any justification from the preceding pattern. Therefore option C is not logically consistent with the sequence.

Option D:
Option D, 1476, coincides exactly with the value computed from the rule for n = 6. It preserves both the powers and coefficients that define the behaviour of the earlier terms. Because this option emerges naturally from the same formula, 1476 is the correct continuation of the number series.