In the number series 5, 42, 191, 578, 1377, the next term is _______. – UGC NET Question

The terms follow the rule aₙ = 2n⁴ + n³ + 2 with n starting at 1. Substituting n = 1, 2, 3, 4 and 5 yields 5, 42, 191, 578 and 1377, exactly the five numbers in the series. Evaluating the same expression for n = 6 gives 2810. Hence 2810 is the only value that fits this established polynomial pattern.

Option A:
Option A, 2786, is 24 less than the value obtained from aₙ = 2n⁴ + n³ + 2 for n = 6. To choose 2786 we would need to reduce the polynomial result only at the last term. Since earlier entries follow the rule without such a change, 2786 cannot correctly continue the series.

Option B:
Option B, 2798, is still below the required value and does not equal the output of the formula for n = 6. It represents a partial adjustment towards 2810 but still breaks the algebraic relationship. Therefore this option does not preserve the exact pattern linking the terms.

Option C:
Option C, 2810, matches precisely the value produced by 2n⁴ + n³ + 2 when n = 6. It maintains the contributions of both the fourth-power and cube terms in exactly the same way as for previous indices. Because the rule explains the given terms and extends naturally to 2810 next, this option is correct.

Option D:
Option D, 2822, is 12 larger than the polynomial value and cannot be generated by the same expression at n = 6. Accepting 2822 would require adding an unexplained increment at the final step. This would break the smooth growth dictated by the formula, so 2822 is not acceptable.