## Factorials: (4n + 1)! / (4n - 1)!

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Mere
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### Factorials: (4n + 1)! / (4n - 1)!

Can someone explain how to do this problem, please?

(4n+1)!/(4n-1)!

stapel_eliz
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Can someone explain how to do this problem, please?

(4n+1)!/(4n-1)!
I'll guess that the instructions were something along the lines of "simplify"...?

To learn how factorials work, try here.

Once you're familiar with how they work, write out the expansions of the two expressions in the fraction. For instance, the numerator would expand as:

. . . . .$1\times 2\times 3\times\, ...\, \times (4n\, -\, 2)(4n\, -\, 1)(4n)(4n\, +\, 1)$

How would the other factorial expression expand? What will cancel out? What then will be left?

Mere
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### Re: Factorials: (4n + 1)! / (4n - 1)!

So then... the denominator would be: 1x2x3x(4n-2)x(4n-1)
And all of that would cancel out with the numerator, leaving (4n)x(4n+1), which is (16(n^2))+4n!

Thank you sooooo much!

stapel_eliz
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So then... the denominator would be: 1x2x3x(4n-2)x(4n-1)
Not quite; you need to include all the numbers in the middle which have been omitted:

. . . . .$1\times 2\times 3\times\, ...\, \times (4n\, -\,2)(4n\, -\, 1)$
And all of that would cancel out with the numerator, leaving (4n)x(4n+1), which is (16(n^2))+4n!
Not quite: 4n(4n + 1) = 4n(4n) + 4n(1) = 16n2 + 4n, not 16n2 + (4n)!.

Make those two corrections, and I believe you've got it!

Mere
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### Re: Factorials: (4n + 1)! / (4n - 1)!

Oh! Okay. Thank you!