Patterns in Pascal's Triangle

Date: 07/21/97 at 14:24:12
From: Camphung Luong
Subject: Pascal's Triangle patterns

I am working on a project about Pascal's triangle. The point of the 
project is to find as many patterns as you can and prove it by 
induction. My partner and I have already proved one pattern. I am 
working on a pattern that the triangle is symmetric, that it's a 
palindrome. I have no clue where to start!  I really need help, Dr. 
Math!

Date: 07/21/97 at 15:41:28
From: Doctor Wilkinson
Subject: Re: Pascal's Triangle patterns

I think if you'll look at how you get from one row of the triangle to 
the next, you can see quite clearly that every row is a palindrome.  
For example, when you go from row 4 to row 5:

    1 4 6 4 1

you get

    1 1+4 4+6 6+4 4+1 1

and the sums are always formed from symmetrically located terms of 
row 4.

Now, how to make a formal proof out of this?

First you need a statement of what it means to be a palindrome.
If you number the terms in row n

 0, 1, 2, ..., n

Then to say it is a palindrome means that

 term 0 is the same as term n
 term 1 is the same as term n-1
 term 2 is the same as term n-2

and so on, which we can express by saying that

 term k is the same as term n-k

So what you want to prove is the identity

 C(n, k) = C(n, n-k)

To show this by induction, you need to show that if it's true for n, 
it is true for n + 1.

And what you have to work with is the "Pascal's triangle identity"

 C(n, k) = C(n-1, k) + C(n-1, k-1)

That should be enough for you.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 07/22/97 at 16:08:25
From: camphung luong
Subject: Patterns in Pascal's Triangle

Hi, Dr. Math.  Thank you for helping me yesterday.  I found out how to 
prove it by induction.  

I need to find some more patterns. Can you help me?  
Thank you again for helping.

Date: 07/22/97 at 16:50:34
From: Doctor Wilkinson
Subject: Re: Patterns in Pascal's Triangle

Good work!

Have you tried adding up the numbers in each row?

How about alternately adding and subtracting the numbers in each row 
(like 1 - 4 + 6 - 4 + 1, for example)?

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/