What is the simplified value of \(\frac{(n+2)!}{(n+1)!}-\frac{n!}{(n-1)!}\)?

Correct answer: 2

Using factorial expansion, \((n+2)!=(n+2)(n+1)!\), so \(\frac{(n+2)!}{(n+1)!}=n+2\). Similarly, \(n!=n(n-1)!\), hence \(\frac{n!}{(n-1)!}=n\). Therefore, the expression is \((n+2)-n=2\). Note that \(n+2\) is only the first ratio, not the difference of the two ratios. Exam tip: In factorial fractions, expand the larger factorial only up to the smaller factorial and cancel common factors.