Question

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

Answer 1

When
`n=1`, you have
`a_1=\frac3{2!}=\frac32`.

When
`n=2`, you have
`a_2=(3^2)/(4!)=\frac38`.

Clearly,
`a_1>a_2`.

Assume
`a_k<a_(k-1)`. Now when
`n=k+1`, you have


`a_(k+1)=(3^(k+1))/((2k+2)!)=\frac3{(2k+2)(2k+1)}*(3^k)/((2k)!)=\frac3{(2k+2)(2k+1)}a_k<a_k`

Therefore by induction the sequence is monotone (decreasing).