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Question on a formula for Pascal's Triangle

  1. Apr 24, 2006 #1
    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?
     
  2. jcsd
  3. Apr 24, 2006 #2

    AKG

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    You start with (n+k)Cr, right? You end up with a sum of terms of the form:

    [tex]\sum _{i = B-A} ^{B} C_i{{n}\choose{D_i}}[/tex]

    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.
     
  4. Apr 24, 2006 #3
    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.
     
  5. Apr 24, 2006 #4

    AKG

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    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?
     
  6. Apr 25, 2006 #5
    The number of terms = k+1 right? Is that Di?
     
  7. Apr 25, 2006 #6

    AKG

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    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:

    [tex]\sum _{i = X} ^Y a_i[/tex]

    How many terms does this sum have?
     
  8. Apr 25, 2006 #7
    Y terms, right?
     
  9. Apr 25, 2006 #8

    AKG

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    Y terms, right?

    No.
     
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