Binomial Expansion

[MrAnderson]

Hi there,
Just wondering if someone can help me in finding the formula "connecting the sum of any k + 1 successive terms, starting at the m th term, in the n+1 th row of Pascal's Triangle which are each multiplied by the binomial coeffcients in the expansion of (a+b)^k ((a+b) the the power of k).

I have attached the investigation is any clarification is needed. You can find my specific question in Part 3 (a).

Thanks,
AC

 

[Deleted member 4993]

Hi there,
Just wondering if someone can help me in finding the formula "connecting the sum of any k + 1 successive terms, starting at the m th term, in the n+1 th row of Pascal's Triangle which are each multiplied by the binomial coeffcients in the expansion of (a+b)^k ((a+b) the the power of k).

I have attached the investigation is any clarification is needed. You can find my specific question in Part 3 (a).

Thanks,
AC

Please refer to replies in:

https://www.freemathhelp.com/forum/...estion-in-the-description.117711/#post-464217

 

[Dr.Peterson]

Thanks for including the original of the problem, and for not making the typo (2 for k) that I pointed out in the other thread.

For those who don't like to open pdf's, here is the problem:


But the question is still ungrammatical! "The formula connecting ..." ought to be followed by two things to be connected, as I read it. The example given helps to make some sense of it; I think "connecting" probably ought to be just "for".

Now, the next question is, what help do you need with this? It's an investigation you are supposed to do, so it would be wrong for us to do the investigating for you. What thoughts have you had, and where are you stuck? Have you, for example, tried evaluating the examples given, or some others, and made a guess at a pattern? Or done some algebraic manipulation?

 

[MrAnderson]

Sorry for not getting back to you sooner. But the formula I found was n+k C k+m-1. I did this by trial and error but am wondering is there a way to algebraically derive this formula?

 

[Dr.Peterson]

They imply that you should be able to figure out the formula not algebraically (though it is probably possible), but by drawing diagrams.

I'm not sure what you mean by "trial and error" -- can you show what you did? It may be part of what they expect you to do initially.

One thing you could do now is to mark up a copy of the triangle with a set of terms and the location of the sum you found. You should see a highly visual meaning for it.

If you hadn't found an answer (or don't want to believe it), you could do as I would do: try some very small examples (2 or 3 terms) and see what the sums are, finding them in the triangle. This should lead to a conjecture, which you can test by some larger examples (4 or 5 terms).

Either way, once you see how the formula relates to the triangle itself, you can think about why it might be true. That's what will make you more confident of your answer.

Tell me what you find.