What is the Value of K?

Date: 01/24/2003 at 07:59:11
From: Kellie
Subject: Values of integers

For positive integer values on N, define N! to be the sum of integers 
1 to N inclusive. For example 3! = 1+2+3 = 6. If 10!-9! = k! and K 
is a positive integer, what is the value of K?

(a)1 (b)4 (c)10 (d)46 (e)55

Date: 01/24/2003 at 09:18:57
From: Doctor Edwin
Subject: Re: Values of integers

Hi, Kellie.

Usually when we write N!, we mean the PRODUCT of the integers 1 to N 
inclusive, not the sum. But let's assume they really meant sum. 

I think the first difficulty here is the way the problem is stated. 
It's in very formal language, which can be a little confusing at 
first.

First let's tackle the question. They're saying, "Sometimes we want to 
add up all the integers between one and n, and do math with that sum. 
Rather than having to write out 1 + 2 + 3 ... etc., over and over, we 
decided to use an exclamation point. So when we say 4!, we mean 1 + 2 
+ 3 + 4, and when we say n!, we mean 1 + 2 + 3 + ... + n." 

Does that make sense? Now, given all that, they give you a problem to 
solve that has two important steps:

   10! - 9! = k!

First step: What's k! ?
Second step: What's k ?

Now that we've talked about the question, let's talk about solving it. 
It helps to write out the values you're adding and subtracting this 
way:

    10!
   - 9!
   ----
     k!

    10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
  -     (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)
  -----------------------------------------  

In that case, what's k! ? Given that k! is:

      1 + 2 + ... + k

can you figure out what k is, given k! ?

Hope this helps. Write back if you're still stuck!

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/