Prove the statement is true using mathematical induction: 2n-1 ≤ n!

Question

Prove the statement is true using mathematical induction:
2n-1 ≤ n!

Answer 1

note the following:

 

n        2n          2n-1

 

0          0            -1

1          2             1

2          4             3

3          6             5

4          8             7

5         10            9

etc.

 

so that for n >= 1, we have that 2n - 1 >= n;  perhaps, this was the result to be proved via Mathematical Induction;  if

 

so, then we would have the following:

 

Theorem:     Sn:  2n - 1 >= n;   n >= 1

 

Proof (via Mathematical Induction):  

 

start with S1:  2(1) - 1 >= 1;  or, 2 - 1 >= 1;  or, 1 >= 1 (which is true)

 

next, show that the assumed truth of Sn implies the truth of S(n+1):  

 

2n - 1 >= n;  or, 2n + 2 - 1 >= n + 2;  or, 2(n+1) - 1 >= n + 2;  but, n+2 >= n+1;  so that, 2(n+1) - 1 >= n + 1; QED