数A1,A2,A3,···,A100满足对于k=1,2,3,4,···,100.Ak比其余99个数的和小k,求A50=( )

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数A1,A2,A3,···,A100满足对于k=1,2,3,4,···,100.Ak比其余99个数的和小k,求A50=( )

数A1,A2,A3,···,A100满足对于k=1,2,3,4,···,100.Ak比其余99个数的和小k,求A50=( )
数A1,A2,A3,···,A100满足对于k=1,2,3,4,···,100.Ak比其余99个数的和小k,求A50=( )

数A1,A2,A3,···,A100满足对于k=1,2,3,4,···,100.Ak比其余99个数的和小k,求A50=( )
根据题意
Ak=(S100-Ak)-k, k=1,2,3,4,···,100
∴2Ak=S100-k
∴Ak=1/2S100-k/2
∴A(k+1)=1/2S100-(k+1)/2
∴A(k+1)-Ak=-1/2
∴{Ak}是等差数列,公差为-1/2

∴A1=1/2S100-1/2, A100=1/2S100-50
根据等差数列求和公式:
∴S100=(A1+A100)*50=(S100-50-1/2)*50
∴49*S100=2525
∴S100=2525/98
∴A50=1/2*S100-25=2525/196-25=-(25×95)/196
=-2375/196

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