What is ((3n+1)!/(3n-1)!) equal to?

Correct answer: (3n+1)(3n)

Use the factorial identity m! = m(m − 1)(m − 2)!, expanding the numerator only until (3n − 1)! appears. We get (3n + 1)! = (3n + 1)(3n)(3n − 1)!. Hence [(3n + 1)(3n)(3n − 1)!]/(3n − 1)! cancels to (3n + 1)(3n). Therefore option A is correct. Option B incorrectly replaces the middle factor 3n with 3n − 1. Option C omits the first factor 3n + 1, and option D includes (3n − 1) as an extra factor even though the corresponding factorial has already been cancelled. The simplification assumes the factorials are defined, as in the usual natural-number domain for n.