MathDB

Suppose an ordered set of

({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}})

real numbers,

n \ge 3

. It is possible to replace the number

{{a} _ {i}}

i = \overline {2, \ n-1}

by the number

a_ {i} ^ {*}

that

{{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}}

. Let

({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}})

be the set with the largest sum of numbers that can be obtained from this, and

({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}})

is a similar set with the least amount. For the odd

n \ge 3

and set

(1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1)

find the values of the expressions

{{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}}

and

{{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}}

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