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!