In the number series 3, 10, 29, 66, 127, the next term is _______. – UGC NET Question

The series follows the rule (a_n = n^3 + 2) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 10, 29, 66 and 127, which match the given terms. For n = 6 the same formula yields (6^3 + 2 = 216 + 2 = 218). Therefore 218 is the unique value that continues the same cubic pattern without any change.

Option A:
Option A equals 218, which is exactly the value of (n^3 + 2) when n = 6. It preserves the single algebraic rule that generates all previous terms. Because no adjustment or exception is needed to obtain 218, this option correctly extends the series.

Option B:
Option B, 214, does not equal (6^3 + 2) and would require subtracting an additional 4 from the formula’s output. This breaks the clean relationship between each term and its index. Hence 214 cannot be accepted as the correct next term.

Option C:
Option C, 222, is larger than the formula’s value and does not arise from (n^3 + 2) for the next integer. Choosing 222 would arbitrarily increase the term and destroy the exact cubic correspondence. Therefore 222 is not a valid continuation of the series.

Option D:
Option D, 226, deviates even further from 218 and again fails to satisfy the expression (n^3 + 2) for n = 6. It would imply altering the rule only at the last step, which is mathematically inconsistent. Thus 226 is not correct.