Question on a formula for Pascal's Triangle

[smallbadwolf]

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?

 

[AKG]

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 C_i and D_i 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.

 

[smallbadwolf]

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.

 

[AKG]

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 C_i and D_i 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?

 

[smallbadwolf]

The number of terms = k+1 right? Is that Di?

 

[AKG]

Yes, number of terms is k+1. Why in the world would that be D_i? D_i 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?

 

[smallbadwolf]

Y terms, right?

 

[AKG]

Y terms, right?

No.