Find an expression for a sequence of numbers ordered within an interval

by L.bronze   Last Updated July 17, 2017 10:20 AM

I have an interval say, $[x_{n-1}, x_n]$. Within this interval, I have two other numbers, say $y_i$ and $z_i$. I want to do some calculations within this given interval. I will give an illustration below. Suppose for $n=1$ I have the following:

(i) $x_0 \lt y_1 \lt z_1 \lt y_2 \lt z_2 \lt x_1$

(ii) $ x_0 \lt z_1 \lt y_1 \lt z_2 \lt y_2 \lt x_1 $

(iii) $ x_0 \lt z_1 \lt z_2 \lt y_1 \lt y_2 \lt x_1 $

(iv) $ x_0 \lt z_1 \lt y_1 \lt y_2 \lt z_2 \lt x_1 $

(v) $ x_0 \lt y_1 \lt z_1 \lt z_2 \lt y_2 \lt x_1 $

Let $k$ be a function. Then in the case of (i), we have

(I) $k(x_0) ( y_1-x_0) + k(y_1) (z_1 - y_1) + k(z_1)(y_2 - z_1) + k(y_2)(z_2 - y_2) + k(z_2) ( x_1 - z_2) $

(II) $ k(x_0) (z_1-x_0) + k(z_1)(y_1 -z_1) + k(y_1)(z_2-y_1) + k(z_2)(y_2-z_2) + k(y_2)(x_1-y_2)~.. $

similarly for (iii), (iv) and (v). How do I write a general formula or expression for such an operation? The numbers $y_i, z_j$ can be ordered arbitrarily between the fixed numbers $x_{n-1}$ and $x_n$.

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