In the number series 5, 58, 273, 836, 2005, the next term is _______. – UGC NET Question

This sequence follows the polynomial rule aₙ = 3n⁴ + n³ + n with n starting from 1. Evaluating the expression for n = 1, 2, 3, 4 and 5 yields 5, 58, 273, 836 and 2005, matching the given terms exactly. For n = 6, we obtain a₆ = 3·6⁴ + 6³ + 6 = 3·1296 + 216 + 6 = 3888 + 222 = 4110. Therefore 4110 is the only number that fits this functional pattern and correctly continues the series.

Option A:
Option A, 4086, is 24 less than the value predicted by 3n⁴ + n³ + n when n = 6. Choosing 4086 would require reducing the correct polynomial result at this term, even though earlier values are exact. This violates the consistency of the pattern, so option A is not acceptable.

Option B:
Option B, 4098, is 12 less than the correct value and again cannot be expressed as 3·6⁴ + 6³ + 6. It provides an approximation but not the true outcome of the formula. In a high-level series problem, such a discrepancy means the option does not represent the genuine rule.

Option C:
Option C, 4122, is 12 greater than the correct term 4110 and likewise does not match the polynomial output. Accepting 4122 would introduce an unexplained upward shift at the last step. Therefore this choice cannot be considered a valid continuation of the series.

Option D:
Option D, 4110, is exactly the value obtained when n = 6 is substituted into 3n⁴ + n³ + n. It preserves the same combination of a strong quartic term, a cubic correction and a linear part that explains the earlier numbers. Because it extends the rule seamlessly, option D is the correct answer.