Patterns in Pascal's TriangleDate: 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/ |
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