Find x_1+x_3+x_5+--------+x_999

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The discussion centers on solving the equation system defined by $x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=...=x_{998}+x_{999}=1$ and $x_1+x_2+x_3+x_4+x_5+...+x_{999}=999$. The goal is to determine the sum of the odd-indexed variables, specifically $x_1+x_3+x_5+x_7+...+x_{999}$. The established conclusion is that the sum of the odd-indexed variables equals 500, derived from the uniform distribution of the variables across the defined constraints.

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Albert1
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$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=--------=x_{998}+x_{999}=1$

$x_1+x_2+x_3+x_4+x_5+--------+x_{999}=999$

find :

$x_1+x_3+x_5+x_7+---+x_{999}=?$
 
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Albert said:
$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=--------=x_{998}+x_{999}=1$

$x_1+x_2+x_3+x_4+x_5+--------+x_{999}=999$

find :

$x_1+x_3+x_5+x_7+---+x_{999}=?$

Given
$x_1+x_2=x_2+x_3=x_3+x_4=x_4+x_5=\cdots=x_{998}+x_{999}=1\cdots(1)$
$x_1+x_2+x_3+x_4+x_5+\cdots+x_{999}=999\cdots(2)$
we have $x_1+x_2=x_2+x_3$ => $x_1= x_3$ ( from (1)
proceeding this way
$x_1 = x_3 = x_ 5 = \cdots = x_{999}\cdots(3)$
and similarly
$x_2 = x_4 = x_ 6 =\cdots = x_{998}\cdots(4)$
further from given relations
$x_1+ x_2 = 1\cdots(5) $ (from(1)
and $500x_1 + 499x_2 = 999\cdots(6) $ from (2), (3) and (4)
multiply (5) by 499 and subtract from (6) giving
$x_1 = 500$
so $x_1+x_3+x_5+x_7+---+x_{999}=500^2 = 250000$
 

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