Question

Question #e0122

Answer 1

Prove for `n=7`
then prove that if true for `n=k-1` then it is true of `n=k`

Explanation:

Part 1
Using pencil and paper arithmetic
`color(white)("XXX")3^7= 2187`
`color(white)("XXX")7! = 5040`
therefore if `n=7`
`color(white)("XXX")3^n < n!`

Part 2
Assume `3^(k-1) < (k-1)!`
`rarr`
`color(white)("XXX")3*3^(k-1) < 3*[(k-1)!]`

`color(white)("XXX")3^k < k*[(k-1)!]color(white)("XXX")`(since `k > 3`)

`color(white)("XXX")3^k < k!`

QED