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

hangainlover

## Homework Statement

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

## The Attempt at a Solution

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

Staff Emeritus
Gold Member
I can find it by plugging it into the calculator
but i want to do it analytically.
Why? Mentor
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 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:
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?

Mentor
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...