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 January 5th, 2017, 09:23 AM #1 Newbie   Joined: Jan 2017 From: Nowhere Posts: 5 Thanks: 1 Find the values of a so that the operation is closed in [0;1] I would appreciate anyone's contribution for the following exercises : 1)Find the values of a so that the operation $\displaystyle x * y = x + y - xy - ax - ay +a$ is stable in the set [0;1] //// First one was solved 2) For a=0, find a formula for $\displaystyle \overbrace{x*x*x*x* .....*x}^{n-times}$. Last edited by MozartInACan; January 5th, 2017 at 09:46 AM.
January 5th, 2017, 03:43 PM   #2
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Quote:
 Originally Posted by MozartInACan I would appreciate anyone's contribution for the following exercises : 1)Find the values of a so that the operation $\displaystyle x * y = x + y - xy - ax - ay +a$ is stable in the set [0;1] //// First one was solved 2) For a=0, find a formula for $\displaystyle \overbrace{x*x*x*x* .....*x}^{n-times}$.
I assume that (2) is related to (1)

$x^n = (-1)^{n-1}x^n + \displaystyle{\sum_{k=1}^{n-1}}~(-1)^{k-1} \begin{pmatrix}n \\ k \end{pmatrix}x^k$

I leave it to you to prove this via induction.

 January 6th, 2017, 08:57 AM #3 Member   Joined: Dec 2016 From: USA Posts: 46 Thanks: 11 When $a = 0$, $\overbrace{x*x*x* \cdots *x}^{n\text{ terms}} = 1 - (1-x)^n$. Last edited by quasi; January 6th, 2017 at 09:06 AM.

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