Question

Prove that n ! > 2 n for n a positive integer greater than or equal to 4. (Note: n! is n factorial and is given by 1 * 2 * ...* (n-1)*n.)

Answer 1

Statement P (n) is defined by  

n! > 2 n

STEP 1: We first show that p (4) is true. Let n = 4 and calculate 4 ! and 2 n and compare them  

4! = 24  

2 4 = 16  

24 is greater than 16 and hence p (4) is true.  

STEP 2: We now assume that p (k) is true  

k! > 2 k

Multiply both sides of the above inequality by k + 1  

k! (k + 1)> 2 k (k + 1)  

The left side is equal to (k + 1)!. For k >, 4, we can write  

k + 1 > 2  

Multiply both sides of the above inequality by 2 k to obtain  

2 k (k + 1) > 2 * 2 k

The above inequality may be written  

2 k (k + 1) > 2 k + 1

We have proved that (k + 1)! > 2 k (k + 1) and 2 k (k + 1) > 2 k + 1 we can now write  

(k + 1)! > 2 k + 1

We have assumed that statement P(k) is true and proved that statment P(k+1) is also true.  

Answer 2

Answer:

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Step-by-step explanation: