Question

Prove that n!<n^n for n≥2​

Answer 1

Explanation:

Hello,

We will prove it by induction.

Step 1 - for n=2

2!=2*1=2

2^2=4

and 2 < 4 so this is true for n=2

Step2 - We assume that this is true for k and we have to prove it for k+1.

Induction hypothesis is
`k!<k^k`

`(k+1)!=(k+1)k!`

We use the induction hypothesis and we we can write that

`(k+1)!=(k+1)k!<(k+1)k^k<=(k+1)(k+1)^k=(k+1)^(k+1)\\\\\text{As }k <= k+1`

so, we prove it for k+1

Step 3- conclusion

for n >= 2 we just proved that
`n!<n^n`