In the number series 7, 50, 191, 514, 1127, the next term is _______. – UGC NET Question

This sequence is described by the rule aₙ = n⁴ + 4n³ + 2 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 into the expression gives 7, 50, 191, 514 and 1127, which are the terms listed in the question. Evaluating the same formula for n = 6 produces 2162. Therefore 2162 is the unique next term that maintains this quartic-plus-cubic structure.

Option A:
Option A, 2146, is 16 less than the value obtained from aₙ = n⁴ + 4n³ + 2 when n = 6. Accepting 2146 would require subtracting 16 from the polynomial outcome only at the sixth term. Since earlier terms agree perfectly with the rule, this option is not compatible with the pattern.

Option B:
Option B, 2150, is 12 less than the correct value and again does not equal the expression’s output for n = 6. It provides a near miss but still disturbs the exact functional relationship linking index and term. Consequently, option B cannot correctly continue the sequence.

Option C:
Option C, 2156, is 6 less than 2162 and likewise fails to match the formula’s value. Choosing 2156 would introduce a small but unjustified deviation from the pattern at the final step. Because the series is exactly captured by aₙ = n⁴ + 4n³ + 2, option C must be rejected.

Option D:
Option D, 2162, coincides exactly with the number generated by the rule for n = 6. It preserves both the fourth-power and cubic components and keeps the constant term unchanged. As it fits seamlessly within the existing algebraic structure, 2162 is the correct next term in the number series.