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## Homework Statement

Prove

**by induction**that

**n!> 2^n for n[tex]\geq[/tex]4**

## Homework Equations

solved example:

P(n):

**2^n>1+3n**let n[tex]\geq[/tex]4

(base): n=4 2^4=16 > 13=1+12=1+(3)(4)

(implication): if for n=k: P(k): 2^k>1+3k, for k[tex]\geq[/tex]4

consider for n=(k+1):

2^(k+1)=2^k*2^1=2^k(1+1)=2^k+2^k >(1+3k) + (1+3k) for k[tex]\geq[/tex]4

>1+3k+3k

[tex]\geq[/tex]1+3k+12 > 1+3k+3

=

**1+3(k+1)**

so P(k) => P(k+1)

## The Attempt at a Solution

(Base) n=k

P(k): k!>2^k

(Implication) show P(k)=> P(k+1)

Consider: n=k+1

(k+1)! > 2^(k+1)

(k+1)! = (k+1)k!> (k+1)*2^k

Please help if you can. I am confused. Thanx.