My Math Forum help with newton binum

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February 27th, 2019, 01:04 PM   #1
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Joined: Dec 2017
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help with newton binum

hello everyone,

I have a few questions about newton's binum.
I was requested to calculates the following:

I have no clue how to calculte either of them
our teacher doesn't explained it
I will be very glad if someone could help me with them
and show me the way of solution

thank you
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 February 28th, 2019, 10:05 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,373 Thanks: 1276 $(a+b)^n = \sum \limits_{k=0}^n \dbinom{n}{k} a^k b^{n-k}$ $\text{Let }b=1$ and see what that gets you. For the second one note that by the nature of Pascal's triangle $\dbinom{n+1}{k} = \dbinom{n}{k-1}+\dbinom{n}{k}$ Write out some of the sum until you see the pattern and simplify it to obtain the term on the right hand side.
 February 28th, 2019, 11:43 AM #3 Newbie   Joined: Dec 2017 From: usa Posts: 4 Thanks: 0 Hey firstly thanks for the rapid answer I miss understood you about the second one I been told that I need to solve it by combinatorics by inventing a problem and solve with it both sides of the euquation; also about the first one when you say comaring b to 1 how should I do it cause I have no idea how to solve that thanks
February 28th, 2019, 12:11 PM   #4
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Joined: Sep 2015
From: USA

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Quote:
 Originally Posted by someone111888 Hey firstly thanks for the rapid answer I miss understood you about the second one I been told that I need to solve it by combinatorics by inventing a problem and solve with it both sides of the euquation; also about the first one when you say comaring b to 1 how should I do it cause I have no idea how to solve that thanks
not comparing b to 1. Letting b=1.

Then you get

$a^n = \sum \limits_{k=1}^n \dbinom{n}{k}a^k$

like for example

$5^n = \sum \limits_{k=0}^n 5^k \dbinom{n}{k}$

Does that term on the right look familiar?

 February 28th, 2019, 12:21 PM #5 Senior Member     Joined: Sep 2015 From: USA Posts: 2,373 Thanks: 1276 this is what you're after re the second problem

 Tags binum, newtin, newton

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