Inequality with summationIf 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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Inequality with summationIf       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      ?

hampiova76 · Accepted Answer

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        .

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)?

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