Please help me prove this simple statement

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In summary, we discussed how to prove a mathematical statement using induction. We also gave examples of a proof by induction, specifically the Bernouilli inequality.
  • #1
tamtam402
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Homework Statement



I'm self-studying Boas book (Mathematical methods in the physical sciences) and one of the problems asks me to prove that for n > 3, n! > 2^n

Homework Equations



There isn't really any "equations".

The Attempt at a Solution



My english is bad so I'll try to explain properly.

I wrote down both series and I can see that with n!, for n > 3 you multiply the previous expression by n to get the next term, but for 2^n every term is simply the last one multiplied by 2. Sorry if the words aren't used correctly but you should see where I'm getting at. However, how do I show this mathematically? Any help would be appreciated!
 
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  • #2
Hi tamtam402! :smile:

Mathematically, we often use induction to show this. Have you seen how to do a proof by induction?
 
  • #3
tamtam402 said:

Homework Statement



I'm self-studying Boas book (Mathematical methods in the physical sciences) and one of the problems asks me to prove that for n > 3, n! > n^2


Homework Equations



There isn't really any "equations".



The Attempt at a Solution



My english is bad so I'll try to explain properly.

I wrote down both series and I can see that with n!, for n > 3 you multiply the previous expression by n to get the next term, but for n^2 every term is simply the last one multiplied by 2. Sorry if the words aren't used correctly but you should see where I'm getting at. However, how do I show this mathematically? Any help would be appreciated!

Your statement that "for n^2 every term is simply the last one multiplied by 2" is false. We do not get 3^2 = 9 by multiplying 2^2 = 4 by 2!

RGV
 
  • #4
Ray Vickson said:
Your statement that "for n^2 every term is simply the last one multiplied by 2" is false. We do not get 3^2 = 9 by multiplying 2^2 = 4 by 2!

RGV

My bad! I meant 2^n. As for proofs by induction, I haven't been taught how to do that. Would you mind giving me an example with my question or something similar if possible? Thanks!
 
  • #5
Check out http://en.wikipedia.org/wiki/Mathematical_induction
Specifically, the sections Description and Example are interesting.

I'll give another example that is more related to your problem. I will try to prove the Bernouilli inequality, which states

[tex](1+x)^n\geq 1+nx[/tex]

for all natural numbers n.

First, we prove the base case, that is we prove the inequality for n=0. This is trivial since

[tex](1+x)^0=1~\text{and}~1+0x=1[/tex]

so the inequality is satisfied for n=0 (and is actually an equality here).

For the induction step, we accept that the theorem is true for n-1 (that is: [itex](1+x)^{n-1}\geq 1+(n-1)x[/itex] holds and is called the induction hypothesis), and we prove it for n.
So, we do:

[tex](1+x)^n=(1+x)^{n-1}(1+x)\geq (1+(n-1)x)(1+x)=1+x+(n-1)x+(n-1)x^2=1+nx+(n-1)x^2\geq 1+nx[/tex]

where we have used the induction hypothesis for the first inequality. The second inequality follows from the fact that [itex](n-1)x^2\geq 0[/itex] always holds.
 
  • #6
One way that you could do is a brute force attempt. n!=n*(n-1)*...*2*1, then examine n!/2^n which gives:
[tex]
\frac{n!}{2^{n}}=\frac{n}{2}\frac{n-1}{2}\frac{n-2}{2}\cdots\frac{3}{2}\frac{2}{2}\frac{1}{2}
[/tex]
Now for what n is this greater than one?
 
  • #7
micromass said:
Check out http://en.wikipedia.org/wiki/Mathematical_induction
Specifically, the sections Description and Example are interesting.

I'll give another example that is more related to your problem. I will try to prove the Bernouilli inequality, which states

[tex](1+x)^n\geq 1+nx[/tex]

for all natural numbers n.

First, we prove the base case, that is we prove the inequality for n=0. This is trivial since

[tex](1+x)^0=1~\text{and}~1+0x=1[/tex]

so the inequality is satisfied for n=0 (and is actually an equality here).

For the induction step, we accept that the theorem is true for n-1 (that is: [itex](1+x)^{n-1}\geq 1+(n-1)x[/itex] holds and is called the induction hypothesis), and we prove it for n.
So, we do:

[tex](1+x)^n=(1+x)^{n-1}(1+x)\geq (1+(n-1)x)(1+x)=1+x+(n-1)x+(n-1)x^2=1+nx+(n-1)x^2\geq 1+nx[/tex]

where we have used the induction hypothesis for the first inequality. The second inequality follows from the fact that [itex](n-1)x^2\geq 0[/itex] always holds.

Thanks, I'm currently reading the wikipedia entry. Shouldn't you have proved this for n+1 instead of n-1 though?
 
  • #8
For the induction step, it doesn't really matter whether you prove it for n, n-1, or n+1, so long as your induction hypothesis is 1 less or you're using strong induction.
 
  • #9
tamtam402 said:
Shouldn't you have proved this for n+1 instead of n-1 though?

It's the same thing if you think about it. I proved that the statement for n-1 implies the statement for n. But this is the same thing as proving that the statement for n implies the statement for n+1.

The idea of induction is "falling dominoes". If you prove that "the first stone will drop" and "if the n'th stone drops then the n+1'th stone drops", then all stones will drop. Indeed, the first stone drops because of the hypothesis. The second stone drops because the first stone drops and the hypothesis. The third stone drops because the second stone drops and the hypothesis,...

But note, if I've written "if the n-1'th stone drops then the n'th stone drops", this would give me the exact same results...
 

1. How do I prove a simple statement?

To prove a simple statement, you need to use logical reasoning and evidence to show that the statement is true. This can include using mathematical equations, scientific experiments, or deductive reasoning.

2. What are some tips for proving a statement?

Some tips for proving a statement include being clear and concise in your argument, using valid evidence to support your claim, and anticipating counterarguments. It's also important to double check your work and make sure your proof is sound.

3. What is the importance of proving a statement?

Proving a statement is important because it allows us to validate the truth of a claim and understand why it is true. This can lead to a deeper understanding of a concept and can also be used to support further research and discoveries.

4. Can you give an example of proving a statement?

Sure, for example, to prove that the sum of two even numbers is always even, you could use the equation 2n + 2m = 2(n+m), where n and m are any even numbers. This shows that the sum of two even numbers can always be expressed as a multiple of 2, thus proving that it is always even.

5. What should I do if I am having trouble proving a statement?

If you are having trouble proving a statement, try breaking it down into smaller, simpler parts and proving each part individually. You can also seek help from a teacher, tutor, or colleague who may have a different perspective or approach that can help you understand and prove the statement.

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