Prove the Binomial theorem by induction.
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1)
(2) Prove the Binomial theorem by induction. The statement of the theorem
is: For all x, y ∈ R,
(x+y) ^n = sum of ( n choose K ) (x)^k (y)^n-k
Hint: Use Pascal’s identity.
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What are your thoughts?
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HallsofIvy
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True, but he's our Subhotosh Kahn!
nwicole, you have posted here often enough now to understand that you need to show something of what you have tried. We need to know what you can do, what you understand about this problem, and where you have a problem in order to know what kind of help you need.
You understand, I hope, that to prove a statement "by induction" you need to show it is true for n= 1 and then show that if it is true for n= k then it is true for n= k+ 1.
So, if n= 1 then you are asked to show that \(\displaystyle (x+ y)^1= \begin{pmatrix}1 \\ 0 \end{pmatrix} x^1 y^0+ \begin{pmatrix}1 \\ 1 \end{pmatrix}x^0y^1\). Is that true? Now suppose the statement is true for n= k. That is, suppose \(\displaystyle (x+ y)^k= \sum_{i= 0}^k \begin{pmatrix}k \\ ix\end{pmatrix} x^{i}y^{k- i}\). Then, prove it for n= k+ 1. Use the fact that \(\displaystyle (x+ y)^{k+1}= (x+ y)(x+ y)^k\).
nwicole, you have posted here often enough now to understand that you need to show something of what you have tried. We need to know what you can do, what you understand about this problem, and where you have a problem in order to know what kind of help you need.
You understand, I hope, that to prove a statement "by induction" you need to show it is true for n= 1 and then show that if it is true for n= k then it is true for n= k+ 1.
So, if n= 1 then you are asked to show that \(\displaystyle (x+ y)^1= \begin{pmatrix}1 \\ 0 \end{pmatrix} x^1 y^0+ \begin{pmatrix}1 \\ 1 \end{pmatrix}x^0y^1\). Is that true? Now suppose the statement is true for n= k. That is, suppose \(\displaystyle (x+ y)^k= \sum_{i= 0}^k \begin{pmatrix}k \\ ix\end{pmatrix} x^{i}y^{k- i}\). Then, prove it for n= k+ 1. Use the fact that \(\displaystyle (x+ y)^{k+1}= (x+ y)(x+ y)^k\).