Question

Proof by induction.

I would like to please have help on how to prove that: 2^n < n! for n > 4

Answer 1

it is true for n = 5 since
\[2^5=32\] whereas
\[5!=120\] now assume true for all
\[k\leq n\] prove it is true for n + 1

Answer 2

that is prove
\[2^{n+1}<(n+1)!\] i.e.
\[2\times 2^n<(n+1)n!\] and since BY INDUCTION
\[2^n<n!\] and
\[2<n+1\] we know
\[2\times 2^n<(n+1)n!\]i.e.
\[2^{n+1}<(n+1)!\]

Answer 3

makes perfect sense :) thank you