Proof by induction: inequality

[olovka]

Hi everyone,
I need some help by proof by mathematical induction. I know what steps it consists of, I understand its basics, and I do not really have problems with equalities and dividable-by-questions, but somehow inequalities are really challenging for me. I am not sure what to do in the last step, after the induction hypothesis.

Here is one that I cannot solve:

(n+1)! > 2^(n+3), for n>=5.

I did the first step, this statement is correct for n=5.

I did the induction hypothesis:

(k+1)! > 2^(k+3), k is element of N.

I tried to solve the last step like this:

(k+2)! > 2^(k+4)

(k+1)! * (k+2) > 2^k * 2^4

And at this point, I have issues. I tried to repleace (k+1)! with 2^(k+3) (though I do not know if I can), but I could not finish it at all. :sad:

Any help and reply will be appreciated, please send one if you know how to solve this problem.

 

[daon2]

Hi everyone,
I need some help by proof by mathematical induction. I know what steps it consists of, I understand its basics, and I do not really have problems with equalities and dividable-by-questions, but somehow inequalities are really challenging for me. I am not sure what to do in the last step, after the induction hypothesis.

Here is one that I cannot solve:

(n+1)! > 2^(n+3), for n>=5.

I did the first step, this statement is correct for n=5.

I did the induction hypothesis:

(k+1)! > 2^(k+3), k is element of N.

I tried to solve the last step like this:

(k+2)! > 2^(k+4)

(k+1)! * (k+2) > 2^k * 2^4

And at this point, I have issues. I tried to repleace (k+1)! with 2^(k+3) (though I do not know if I can), but I could not finish it at all. :sad:

Any help and reply will be appreciated, please send one if you know how to solve this problem.

((k+1)+1)! = (k+2)(k+1)! > (k+2)2^(k+3) by the Induction Hypothesis

You also know that (k+2)>2 (since k>=5) so...

 

[olovka]

((k+1)+1)! = (k+2)(k+1)! > (k+2)2^(k+3) by the Induction Hypothesis

You also know that (k+2)>2 (since k>=5) so...

Thank you for your answer, but unfortunately still I do not really get it. In the first line, why is it possible to multiply the right hand side by (k+2) ? In the induction hypothesis we stated (k+1)! > 2^(k+3). How could we make from the right hand side (k+2)2^(k+3) ?

Thank you in advance for your help, I hope I will finally understand what is going on here.