Question

show by induction that

Answer 1

\[n!<n^n\]

Answer 2

assuming
k! < k^k .....(1)

we have to prove
(k+1)! < (k+1)^(k+1)
(k+1)(k!) <(k + 1)^k * (k+1)

since k+1 >0
we see
k! < (k+1)^k ...(2)
binomial expansion will show RHS = k^k + something positive
from (1) , we easily see (2) is true..

Answer 3

hope you didnt mind i missed the initial step of induction..

Answer 4

no i dont mind cos i had already done that ,it is great