In the number series 3, 17, 55, 129, 251, the next term is _______. – UGC NET Question

This series is based on the formula aₙ = 2n³+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2·1³+1 = 3, 2·2³+1 = 17, 2·3³+1 = 55, 2·4³+1 = 129 and 2·5³+1 = 251. For n = 6 the expression gives 2·6³+1 = 432+1 = 433. Therefore 433 is the next term that keeps the cubic pattern intact.

Option A:
Option A, 413, is 20 less than the value provided by 2·6³+1. It does not satisfy the generating formula and would require an unexplained downward adjustment. Hence 413 cannot be considered the correct continuation of the series.

Option B:
Option B, 421, is closer but still does not match 2·6³+1, so it fails to maintain the exact relationship between n and aₙ. Choosing it would make the pattern inconsistent with the earlier terms. Thus 421 is not a valid answer.

Option C:
Option C, 433, is exactly the value given by the rule aₙ = 2n³+1 when n = 6. It carries the same structure forward without alteration and fits perfectly with all preceding data. For this reason, 433 is the correct next term.

Option D:
Option D, 448, is larger than the formula’s output and cannot be obtained from 2n³+1 for n = 6. Adopting 448 would abandon the precise cubic relationship that defines the sequence, so it is not correct.