Question on a formula for Pascal's Triangle

AI Thread Summary
The discussion revolves around finding a formula that connects (k+1) successive coefficients in the nth row of Pascal's Triangle to a coefficient in the (n+k)th row. The user explores this by selecting specific values for n and k, illustrating their findings with binomial coefficient notation. They identify a pattern in the coefficients but struggle to express it in a formal equation. Key variables like A, B, Ci, and Di are introduced to help articulate the relationship, with confusion arising over their definitions and how they relate to n, k, and r. The conversation emphasizes the need for clarity in mathematical notation and the relationships between these variables.
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I'm doing an investigation on Binomial Coefficients in my HL math class, and the problem reads:

"There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Find this formula."

Basically, what I did first was I chose arbitrary values of n and k to start with, n being the row number and k being the kth number in that row (confusing, I know). So I chose n = 4 and k = 3.

So in row 4 I saw that 4 (3+1=4) successive coefficients are 4, 6, 4, 1. Now I can see that in the 7th row ((n+k)th row = (4+3)th row = 7th row ) I have to relate the 35 (7C5 is the nCr notation for it) to those successive coefficients 4, 6, 4, and 1 in the 4th row.

I did this using the nCr notation.

7C5 = 6C4 + 6C5

= 5C3 + 5C4 + 5C4 + 5C5

= 4C2 + 4C3 + 4C3 + 4C4 + 4C3 + 4C4 +4C4 + 4C5 (I stop here because I've worked my way from the (n+k)th row to the nth row, or the 4th row in this case.)

= (1) 4C2 + (3) 4C3 + (3) 4C4 + (1) 4C5

These coefficients in front of the row 4 coefficients are the same coefficients that appear in row 3 of Pascal's Triangle. I see the pattern, I just can't seem to put it in formulaic form.

Since I typed a long process and showed my work, I'll type the question again:

"There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Find this formula."

Can anyone help me put my work into a formula? Can anyone find this formula?
 
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You start with (n+k)Cr, right? You end up with a sum of terms of the form:

\sum _{i = B-A} ^{B} C_i{{n}\choose{D_i}}

right? Figure out how A, B, and the Ci and Di depend on n, k, and r. For starters, A depends only on k. That is to say that the number of terms in the sum depends only on k.
 
I'm a bit confused. What is the difference between r and k? I thought they were the same--k = the kth or the rth number in the row n. Also, how can you introduce new variables such as Ci, Di, A and B when I only started with n and k? I'm sorry I'm not very skilled in math but how would I figure out how these relate to n, k and r? Argh I'm confused.
 
You started with (n+k)Cr. In your example, you start with 7C5, where you took n=4, k=3, and r=5. A, B, and the Ci and Di can be expressed in terms of n, k, and r.

Okay, start with something simple. You start with one binomial coefficient, and end up with a sum of terms. How many terms do you end up with, and how does this (the number of terms) depend on k?
 
The number of terms = k+1 right? Is that Di?
 
Yes, number of terms is k+1. Why in the world would that be Di? Di is suggestively labelled, i.e. it tells you that it depends on i. Okay, look at the following:

\sum _{i = X} ^Y a_i

How many terms does this sum have?
 
Y terms, right?
 
Y terms, right?

No.
 
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