设数列a1,a2,a3...,an,...中的每一项都不为0.证明:{an}为等差数列的充分必要条件是:对任何n属于N,都有1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1
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![设数列a1,a2,a3...,an,...中的每一项都不为0.证明:{an}为等差数列的充分必要条件是:对任何n属于N,都有1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1](/uploads/image/z/14562582-6-2.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97a1%2Ca2%2Ca3...%2Can%2C...%E4%B8%AD%E7%9A%84%E6%AF%8F%E4%B8%80%E9%A1%B9%E9%83%BD%E4%B8%8D%E4%B8%BA0.%E8%AF%81%E6%98%8E%EF%BC%9A%7Ban%7D%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E7%9A%84%E5%85%85%E5%88%86%E5%BF%85%E8%A6%81%E6%9D%A1%E4%BB%B6%E6%98%AF%EF%BC%9A%E5%AF%B9%E4%BB%BB%E4%BD%95n%E5%B1%9E%E4%BA%8EN%2C%E9%83%BD%E6%9C%891%2Fa1%2Aa2%2B1%2Fa2%2Aa3%2B...1%2Fan%2Aan%2B1%3Dn%2Fa1%2Aan%2B1)
设数列a1,a2,a3...,an,...中的每一项都不为0.证明:{an}为等差数列的充分必要条件是:对任何n属于N,都有1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1
设数列a1,a2,a3...,an,...中的每一项都不为0.证明:{an}为等差数列的充分必要条件是:对任何n属于N,都有1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1
设数列a1,a2,a3...,an,...中的每一项都不为0.证明:{an}为等差数列的充分必要条件是:对任何n属于N,都有1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1
先证必要性
若为等差数列,则a1=a 差为d
1/(a1a2)+1/(a2a3)+……1/(anan+1)=1/a(a+d)+1/(a+d)(a+2d)+……1/(a+(n-1)d)(a+nd)
裂项得=(1/d)*(1/a-1/(a+d)+1/(a+d)……-1/(a+(n-1)d)+1/(a+(n-1)d)-1/(a+nd)=(1/d)*(1/a-1/(a+nd))=n/a(a+nd)=n/a1*an+1
再证充分性由
1/a1*a2+1/a2*a3+...1/an*an+1=n/a1*an+1
1/a1*a2+1/a2*a3+...1/an-1*an=n/a1*an
两式相减得
1/(an*an+1)=n/a1*an+1-n/a1*an
-->nan-(n-1)an+1=a
又可得(n-1)an-1-(n-2)an=a
再两式相减得(2n-2)an-(n-1)(an-1+an+1)=0
2an-an-1-an+1=0
-->an+1-an=an-an-1得证