In the number series 2, 3, 7, 25, 121, the next term is _______. – UGC NET Question

Each term of this series can be expressed as n!+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1!+1 = 2, 2!+1 = 3, 3!+1 = 7, 4!+1 = 25 and 5!+1 = 121. For n = 6 the same factorial rule gives 6!+1 = 720+1 = 721. Therefore 721 is the only value that keeps the factorial-based pattern intact.

Option A:
Option A, 701, is much smaller than 6!+1 and does not correspond to n!+1 for any integer n following 5. It breaks the rule that clearly generates the given terms. Thus 701 cannot be the correct next term in the sequence.

Option B:
Option B, 709, is closer numerically but still fails to equal 6!+1. It is not of the form n!+1 for the next natural number and would therefore require changing the underlying idea of the series. Hence 709 is not appropriate.

Option C:
Option C, 721, exactly matches the value obtained when we compute 6!+1. It preserves the simple and elegant rule that each term is one more than the factorial of its index. For this reason, 721 is the correct continuation of the series.

Option D:
Option D, 736, exceeds the correct factorial-based value and likewise cannot be written as n!+1 for the required n. Choosing 736 would abandon the factorial structure, so it is not a valid answer.