Thread: Inequalities

Can anyone help me confirm if I've got this one correct? Particularly the part at A, where I've defined (k + 2)! = (k + 1)!(k + 2).

Many thanks.

Q. (n + 1)! > 2ⁿ, for n

Attempt: Step 1: For n = 1...
(1 + 1)! = 2! = 2 & 2¹ = 2
Since 2 > 2, the statement is true for n = 1.

Step2: Assume the statement is true for n = k, i.e. assume (k + 1)! > 2^k.
We must now show that the statement is true for n = k + 1,
i.e. ((k + 1) + 1)! > 2^k+1 if (k + 2)! > 2^k+1
A: (k + 2)! = (k + 1)!(k + 2) > (k + 1)2^k...((k + 1)! > 2^k...assumed)
If (k + 1)2^k > 2^k+1, then (k + 2)! > 2^k+1
if (k + 1)2^k > 2^k+1
if (k + 1)2^k - 2^k+1 > 0
if 2^k[(k + 1) - 2] > 0...which is true for k > 1
=> (k + 2)! > 2^k+1
Therefore, the statement is true for n = k + 1, if true for n = k. Thus, the statement is true for all n > 2, n

Originally Posted by GrigOrig99

Can anyone help me confirm if I've got this one correct? Particularly the part at A, where I've defined (k + 2)! = (k + 1)!(k + 2).
Many thanks. Q. (n + 1)! > 2ⁿ, for n
Attempt: Step 1: For n = 1...
(1 + 1)! = 2! = 2 & 2¹ = 2
Since 2 > 2, the statement is true for n = 1.
Step2: Assume the statement is true for n = k, i.e. assume (k + 1)! > 2^k.
We must now show that the statement is true for n = k + 1,
i.e. ((k + 1) + 1)! > 2^k+1 if (k + 2)! > 2^k+1
A: (k + 2)! = (k + 1)!(k + 2) > (k + 1)2^k...((k + 1)! > 2^k...assumed)
If (k + 1)2^k > 2^k+1, then (k + 2)! > 2^k+1
if (k + 1)2^k > 2^k+1
if (k + 1)2^k - 2^k+1 > 0
if 2^k[(k + 1) - 2] > 0...which is true for k > 1
=> (k + 2)! > 2^k+1
Therefore, the statement is true for n = k + 1, if true for n = k. Thus, the statement is true for all n > 2, n

You tend to make too much of this.
Part of step 2 should note that assuming we have . So you kn ow that .
Look at

Ok, so A should be rewritten as: (k + 2)! = (k + 1)!(k + 2) > (k + 2)2^k...?

Originally Posted by GrigOrig99

Ok, so A should be rewritten as: (k + 2)! = (k + 1)!(k + 2) > (k + 2)2^k...?

Yes.

Cont. from there:
If (k + 2)2^k > 2^k+1, then (k + 2)! > 2^k+1
(k + 2)2^k > 2^k+1
if (k + 2)2^k - 2^k+1 > 0
if 2^k[(k + 2) - 2^k] > 0...true when k > 1
=> (k + 2)! > 2^k+1