Question

Prove that sequence an = (3^n)/(2n)! is monotone

Answer 1

When $n=1$, you have $a_1=\dfrac3{2!}=\dfrac32$.

When $n=2$, you have $a_2=\dfrac{3^2}{4!}=\dfrac38$.

Clearly, $a_1>a_2$.

Assume $a_k$. Now when $n=k+1$, you have

$a_{k+1}=\dfrac{3^{k+1}}{(2k+2)!}=\dfrac3{(2k+2)(2k+1)}\times\dfrac{3^k}{(2k)!}=\dfrac3{(2k+2)(2k+1)}a_k$

Therefore by induction the sequence is monotone (decreasing).