(n+1) <100000 how do i find n?

[hangainlover]

Homework Statement

(n+1)! <100000 how do i find n?

Homework Equations

The Attempt at a Solution

I can find it by plugging it into the calculator
but i want to do it analytically.
Thanks

 

[nickjer]

Not sure of an exact solution, but an quick and easy approximation is to use Stirling's approximation: http://en.wikipedia.org/wiki/Stirling's_approximation

 

[Hurkyl]

I can find it by plugging it into the calculator
but i want to do it analytically.

Why?

 

[Mark44]

I agree with Hurkyl. Just do it by trial and error using a calculator.

 

[icystrike]

You should be expecting something really small for n.
Sometimes you have to take a little time to meditate on the question before you start(considering what approach you want to take and by your intuition will it lead you to anywhere).

Thus, if you work it out ,

0! = 1 By def^{n}
1! = 1
2! = 2
3! = 3x2 = 6
4! = 4x3x2 = 24 (realizing that its slightly smaller than the factor 25 of 10,000)
5! = 5x4x3x2 = 5x24 < 400x25 = 10,000
6! = 6x5x4x3x2 = 30x24 < 400x25 = 10,000
7! = 210x24 < 400x25 = 10,000
8! = 1680 x 24 ( you realize that one of the factor is way larger than 400 and another is smaller than 24 , inconclusive)

Consider using a factor larger than 25 ... how about 26? 2x13 .. we have yet to reach 13! , analogously you will choose 28 as your choice.

8! = 4x360x28 > 2x200x25 = 10,000 Now you may draw a conclusion.

 

[dacruick]

divide both sides by a factorial. 100000/ ! = n +1. n = 100000/! - 1.

 

[icystrike]

divide both sides by a factorial. 100000/ ! = n +1. n = 100000/! - 1.

What operation is that?

 

[Mark44]

divide both sides by a factorial. 100000/ ! = n +1. n = 100000/! - 1.

This makes no sense at all.

 

[fedderenator]

LMAO divide both sides by a factorial...wow...