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

1. Mar 16, 2010

hangainlover

1. The problem statement, all variables and given/known data
(n+1)! <100000 how do i find n?

2. Relevant equations

3. The attempt at a solution
I can find it by plugging it into the calculator
but i want to do it analytically.
Thanks

2. Mar 16, 2010

nickjer

3. Mar 16, 2010

Hurkyl

Staff Emeritus
Why?

4. Mar 17, 2010

Staff: Mentor

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

5. Mar 17, 2010

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 realise 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.

Last edited: Mar 17, 2010
6. Mar 17, 2010

dacruick

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

7. Mar 17, 2010

icystrike

What operation is that?

8. Mar 17, 2010

Staff: Mentor

This makes no sense at all.

9. Mar 17, 2010

fedderenator

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

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