In the number series 5, 50, 219, 644, 1505, the next term is _______. – UGC NET Question

The terms follow the rule aₙ = 2n⁴ + 2n³ + n with n starting from 1. For n = 1, 2, 3, 4 and 5 this formula produces 5, 50, 219, 644 and 1505, matching the given series perfectly. Evaluating the same expression for n = 6 yields 3030. Hence 3030 is the only number that preserves this quartic-plus-cubic pattern.

Option A:
Option A, 3006, is 24 less than the value obtained from aₙ = 2n⁴ + 2n³ + n at n = 6. To accept 3006 we would need to reduce the polynomial result arbitrarily at the sixth term. This change is not reflected anywhere else in the progression, so option A is incorrect.

Option B:
Option B, 3020, is 10 less than the correct value and also fails to equal the formula’s output for n = 6. Although numerically close, it breaks the precise algebraic relationship between n and aₙ. Therefore option B does not correctly continue the series.

Option C:
Option C, 3030, matches exactly the value produced by the expression when n = 6. It maintains the same contributions from the fourth-power and cubic terms along with the linear part as in earlier terms. Because this rule explains the entire sequence and extends naturally to 3030, option C is correct.

Option D:
Option D, 3046, is 16 greater than the expected value and cannot be derived from the formula at n = 6. Choosing 3046 would introduce an unjustified increase at the last step, disrupting the pattern. Thus option D is not a valid continuation.