Are these factorial statements accurate?

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Discussion Overview

The discussion centers around the accuracy of various factorial statements and their properties, including specific examples and generalizations. The scope includes mathematical reasoning and exploration of factorial definitions.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant states that 6! = 6*5*4*3*2*1 is correct.
  • Another participant confirms the correctness of the statements regarding (n+2)!, (2n+2)!, and (500n+3)!, asserting they follow the factorial definition.
  • A later reply suggests that using these properties allows for simplifications, such as \(\frac{(n+2)!}{n!} = (n+1)(n+2)\).
  • Another participant proposes a generalization: \(\frac{(n+a)!}{n!}=\prod_{i=1}^a n+i\).

Areas of Agreement / Disagreement

Participants generally agree on the correctness of the factorial statements presented, but there is no explicit consensus on the implications or applications of these properties.

Contextual Notes

The discussion does not address potential limitations or assumptions underlying the factorial definitions or the generalizations proposed.

Dannbr
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so,

6! = 6*5*4*3*2*1

(n+2)! = (n+2)(n+1)(n)!

(2n+2)! = (2n+2)(2n+1)(2n)!

(500n+3)! = (500n+3)(500n+2)(500n+1)(500n)!

Are all these statements correct?
 
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Dannbr said:
so,

6! = 6*5*4*3*2*1

(n+2)! = (n+2)(n+1)(n)!

(2n+2)! = (2n+2)(2n+1)(2n)!

(500n+3)! = (500n+3)(500n+2)(500n+1)(500n)!

Are all these statements correct?

Yes.
 
Dannbr said:
so,

6! = 6*5*4*3*2*1

(n+2)! = (n+2)(n+1)(n)!

(2n+2)! = (2n+2)(2n+1)(2n)!

(500n+3)! = (500n+3)(500n+2)(500n+1)(500n)!

Are all these statements correct?

Yes. Using these properties can allow you to simplify something like:

[tex]\frac{(n+2)!}{n!} = (n+1)(n+2)[/tex]
 
You could generalize to get [tex]\frac{(n+a)!}{n!}=\prod_{i=1}^a n+i[/tex]
 

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