Inequality with summation If ai positive numbers and n>=2 (the subscripts are taken modulo n), how can I prove the following inequality n sum_(k=1)^n 1/((n-1)a_k+a_(k+1))<=sum_(k=1)^n 1/(a_k)?

wijii4 2022-09-24 Answered
Inequality with summation
If a i positive numbers and n 2 (the subscripts are taken modulo n), how can I prove the following inequality
n k = 1 n 1 ( n 1 ) a k + a k + 1 k = 1 n 1 a k ?
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Answers (1)

hampiova76
Answered 2022-09-25 Author has 5 answers
By C-S
n k = 1 n 1 ( n 1 ) a k + a k + 1 n ( n 1 + 1 ) 2 k = 1 n ( ( n 1 ) 2 ( n 1 ) a k + 1 2 a k + 1 ) =
= 1 n k = 1 n ( n 1 a k + 1 a k + 1 ) = 1 n k = 1 n ( n 1 a k + 1 a k ) = k = 1 n 1 a k .
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