Formula for Connection between Rows of Pascal's Triangle
Date: 11/15/2003 at 22:25:29 From: Michelle Subject: connection between the rows in the Pascal Triangle I've been given this problem, and I'm not sure how to do it: There is a formula connecting any (k+1) coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Find this formula. Is k just a variable or does it represent an actual part of the triangle? Any help you can give me with this problem would be appreciated.
Date: 11/17/2003 at 19:10:11 From: Doctor Schwa Subject: Re: connection between the rows in the Pascal Triangle Hi Michelle, Yes, k is a variable, or more precisely a parameter. Let's say k is 3. Then you want to look at k+1, or 4, numbers next to each other in the nth row. For example, let's look at the 4th row. One possible set of 4 numbers next to each other from the 4th row is 4 6 4 1 in that order. They are saying that those four numbers are related to one number in the (4+3), or 7th row, where you have 1 7 21 35 35 21 7 1. Can you see any relationship? Now, test your theory: look at, say, 5 10 10 5 from the 5th row and see if it relates to something in the (5+3) or 8th row in the same way as your idea for the 4th and 7th row. Rather than start with k = 3, though, you'll find it easier to start with k = 1: look for a pattern that relates two numbers in the nth row to one number in the (n+1)th row. That should be straightforward. Then try k = 2: is there a pattern that relates three consecutive numbers in the nth row with one number in the (n+2)th row? This one is a bit trickier, but once you see it, you'll have the key to the whole thing! It also sometimes helps to think in terms of combinations (choosing elements of a set, for instance) ... Does that help? - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
Date: 11/24/2003 at 23:03:53 From: Michelle Subject: Thank you (connection between the rows in the Pascal Triangle) Thank you for answering my question. I did see a pattern after a while...thanks again!
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