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February 27th, 2019, 01:04 PM   #1
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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:


http://www.interload.co.il/upload/9048222.png

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
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$(a+b)^n = \sum \limits_{k=0}^n \dbinom{n}{k} a^k b^{n-k}$

$\text{Let }b=1$ 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.
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February 28th, 2019, 11:43 AM   #3
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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
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February 28th, 2019, 12:11 PM   #4
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Quote:
Originally Posted by someone111888 View Post
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?
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February 28th, 2019, 12:21 PM   #5
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this

is what you're after re the second problem
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