Factoring (n+1)! - 1 + (n+1)(n+1)! Step-by-Step

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    Factorials
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Discussion Overview

The discussion revolves around the factoring of the expression (n+1)! - 1 + (n+1)(n+1)!, exploring the steps involved in reaching a simplified form. Participants engage in a mix of mathematical reasoning and clarification regarding factorials and their properties.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests a step-by-step explanation for factoring the expression (n+1)! - 1 + (n+1)(n+1)!
  • Another participant challenges a previous claim, suggesting it incorrectly implies (n+1)! = 1 for all n.
  • A participant proposes a factoring approach, suggesting that (n+1)! + (n+1)(n+1)! can be expressed as (n+1)!*(1 + n + 1) - 1.
  • There is confusion about how the expression simplifies to (1 + n + 1), with a participant questioning the relationship between (n+1)! and its factorial expansion.
  • Participants discuss the possibility of substituting variables to simplify the factoring process, with one suggesting that substituting (n+1)! with another variable might help.
  • Another participant introduces a different expression involving k and asks for factoring guidance, indicating a potential shift in focus from factorials.
  • There is a suggestion that substituting variables could be beneficial, but it is emphasized that the original variables should be reintroduced for clarity.
  • A participant confirms understanding of how the initial problem relates to (n+2)! - 1, indicating some progress in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of certain claims, particularly regarding the implications of factorial properties. The discussion remains unresolved with multiple competing perspectives on the factoring process and the use of variable substitution.

Contextual Notes

Some participants express confusion about the steps involved in factoring, and there are unresolved questions about the relationships between factorials and their expansions. The discussion also touches on the applicability of variable substitution in simplifying expressions.

L²Cc
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can you please explain (step by step) how to factor the following:

(n+1)! - 1 + (n+1)(n+1)!

i have the answer, don't know how to get there!
 
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radou, that's absolutely wrong. You realized you just proved (n+1)!=1 for all n?
 
Office_Shredder said:
radou, that's absolutely wrong. You realized you just proved (n+1)!=1 for all n?

Yes, I just did. :biggrin:
 
That's not fair! You can't delete your post like that! :P

Getting back on topic:

(n+1)! + (n+1)*(n+1)! - 1 = (n+1)!*(1 + n + 1) - 1.

Can you go from there?
 
thanks for the quick reply...
You see that's where i get confused...how did you end up with (1 + n + 1)...
Is (n+1)! = (n-1)(n)(n+1)...and so forth?!
 
(n+1)! + (n+1)*(n+1)! = (n+1)!*1 + (n+1)!*(n+1). You factor (n+1)! out and are left with 1 + n + 1

And yes, (n+1)! = (n+1)*n*(n-1)...
 
does the factoring process of (n+1)! involve (n+1)! = (n+1)*n*(n-1)... ?
 
[tex](n+1)! = n!(n+1)[/tex]
 
oh all right i see what you guys are coming at...would it have been easier if i had substituted any variable (say, h) for (n+1)!...? and then factored it...
Btw, this is part of a mathematical induction...im trying to understand factorials better!
thank you guys!
 
  • #10
(this does not involve factorials anymore)...
[k(k+1)(k+2)(k+3) + 4(k+1)(k+2)(k+3)]/4
factor this out...
What's the common factor? How did you get there? (ok i hope it doesn't require expanding the polynomials :p)
Again, would it be easier if i substituted every (k+x) by a different variable, where (k+1) would equal to variable 'A', (k+2) = B, and so forth?
 
Last edited:
  • #11
L²Cc said:
Again, would it be easier if i substituted every (k+x) by a different variable, where (k+1) would equal to variable 'A', (k+2) = B, and so forth?

If it helps you, sure.
 
  • #12
L²Cc said:
Again, would it be easier if i substituted every (k+x) by a different variable, where (k+1) would equal to variable 'A', (k+2) = B, and so forth?

Only if afterwards you plug the (k+x)'s back in, so you can see what your new thing looks like.

And I disagree, the problem you posted does deal with factorials.


Just to confirm, you did figure out how the first problem became (n+2)! - 1 right?
 

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