I'm doing an investigation on Binomial Coefficients in my HL math class, and the problem reads:(adsbygoogle = window.adsbygoogle || []).push({});

"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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# Homework Help: Question on a formula for Pascal's Triangle

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