In the number series 4, 42, 272, 1090, 3252, the next term is _______. – UGC NET Question

The terms of this sequence are generated by aₙ = n⁵ + n³ + 2 with n beginning at 1. Substituting n = 1, 2, 3, 4 and 5 gives 4, 42, 272, 1090 and 3252, which agrees perfectly with the series. For n = 6 we obtain a₆ = 6⁵ + 6³ + 2 = 7776 + 216 + 2 = 7994. Hence 7994 is the unique value that preserves this quintic-plus-cubic pattern.

Option A:
Option A, 7970, is 24 less than the formula’s output at n = 6. To accept 7970 we would have to subtract 24 from the correct result only at the sixth term, even though earlier values match the formula exactly. This breaks the algebraic consistency of the series, so option A is incorrect.

Option B:
Option B, 7982, is 12 smaller than the correct value 7994 and cannot be produced by n⁵ + n³ + 2 for n = 6. It represents a near miss rather than a true continuation of the rule. Since the pattern is purely functional in n, any such deviation must be rejected.

Option C:
Option C, 7994, is exactly the result of substituting n = 6 into the expression n⁵ + n³ + 2. It maintains the same structure in which the fifth-power term dominates and the cubic and constant terms supply additional increments. Because it fits all given data and extends the rule without modification, this option is correct.

Option D:
Option D, 8006, is 12 greater than the computed value and again cannot arise from the same polynomial at n = 6. Accepting 8006 would introduce an unjustified extra amount at the final step. Therefore this option does not correctly extend the number series.