In the number series 6, 13, 32, 69, 130, the next term is _______. – UGC NET Question

This sequence is generated by the formula aₙ = n³+5 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 yields 1³+5 = 6, 2³+5 = 13, 3³+5 = 32, 4³+5 = 69 and 5³+5 = 130. For n = 6 the same expression gives 6³+5 = 216+5 = 221. Therefore 221 is the correct continuation that keeps the series consistent.

Option A:
Option A, 221, comes directly from the rule aₙ = n³+5 when n = 6. It preserves the cubic-plus-constant pattern that perfectly explains all previous terms. Because no modification of the rule is needed, this option fits seamlessly as the next term.

Option B:
Option B, 211, does not equal 6³+5 and is obtained by subtracting an unrelated amount from the correct value. This breaks the neat pattern connecting the terms to their indices. Thus 211 cannot be accepted as the correct next term.

Option C:
Option C, 215, is closer to 221 but still does not agree with the generating formula. It would require changing the constant 5 at the end, which is consistently applied to earlier terms. Hence 215 is not a valid continuation.

Option D:
Option D, 229, overshoots the correct cubic value and again fails to match the rule aₙ = n³+5. Selecting 229 would destroy the simple uniform structure of the sequence, so it is not the right answer.