Question

Prove, by process of Mathematical Induction, the inequality for the indicated integer values of n.

n!>2^n, n≥4

Answer 1

I have solved till a point and do not see how to proceed further...


1.) First we check if it is true/valid for n=4 and n=5.

We find it is true because

for n=4, n!=4!=4*3*2*1=24 and 2^4=16

for n=5, n!=5!=5*4*3*2*1=120 and 2^5=32

So, n!>2^n is true in both cases.

2.) Now, assuming that k!>2^k

we need to show that (k+1)!>2^(k+1)

How do I do the 2.) part???

Answer 2

You know that \((k+1)!=k!(k+1)\) and \(2^{k+1}=2\cdot2^{k}\). Since \(k+1>2\), and \(k!>2^k\) by the inductive hypothesis, you know that \((k+1)!>2^{k+1}\).