8n divides (4n) Proof by induction

In summary, induction is a proof technique used in mathematics to prove a statement for all integers starting from a base case, and then extending it to the next integer. In this proof, the base case is n=1 and the induction step involves assuming the statement holds for n=k and then showing it also holds for n=k+1. This proof only works for positive integers and values of n that are divisible by 4. It is an important proof in mathematics as it demonstrates the power of induction and the relationship between divisibility and multiplication.
  • #1
kathrynag
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0

Homework Statement


I need to do this by proof by induction
8n divides (4n)!

Homework Equations





The Attempt at a Solution


I already did it for 1, now I need to do for k and k+1
(4k)!t=8k
 
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  • #2
If 8k divides (4k)!, how can 8k be a multiple of (4k)! as you wrote?
 
  • #3
[tex]8^{k}[/tex] divides (4k)!
[tex]8^{k}[/tex]*t=(4k)!
 
  • #4
Simplify (4(k + 1))! in terms of (4k)!
 
  • #5
(4k+4)!=(4k+4)(4k+3)(4k+2)(4k+1)(4k)!
 
  • #6
You already know that [tex]8^k[/tex] divides (4k)!, so does 8 divide everything else?
 
  • #7
Yes...
 
  • #8
Then you have all the details for the complete proof.
 
  • #9
Ok, I see...
 

1. How does induction work in this proof?

Induction is a mathematical proof technique that involves proving a statement for all integers starting from a base case and then using the fact that if the statement holds for a certain integer, it also holds for the next integer. In this particular proof, we will start with the base case of n=1 and then show that if the statement is true for n=1, it is also true for n+1.

2. What is the base case for this proof?

The base case for this proof is when n=1. We will show that 8 divides 4, which is true because 4=8*0. Therefore, the statement holds for n=1.

3. How do we prove the induction step?

To prove the induction step, we assume that the statement holds for n=k and then show that it also holds for n=k+1. In this proof, we will assume that 8k divides 4k and then use algebraic manipulation to show that 8(k+1) divides 4(k+1). This will prove that if the statement is true for n=k, it is also true for n=k+1.

4. Can we use any value for n in this proof?

No, this proof only works for positive integers. Additionally, the statement will only hold for values of n that are divisible by 4. For example, n=2 will not work because 8 does not divide 8.

5. Why is this proof important in mathematics?

This proof is important because it demonstrates the power of mathematical induction in proving statements for all integers. It also shows the relationship between divisibility and multiplication, which is a fundamental concept in mathematics. Induction is a widely used proof technique in many areas of mathematics and this proof serves as a good example of its application.

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