【急】已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列已知数列an满足1/(a1a2)+1/(a2a3)+……1/(an-1an)=(n-1)/a1an,求证为等差数列.
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![【急】已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列已知数列an满足1/(a1a2)+1/(a2a3)+……1/(an-1an)=(n-1)/a1an,求证为等差数列.](/uploads/image/z/7122397-13-7.jpg?t=%E3%80%90%E6%80%A5%E3%80%91%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%E6%BB%A1%E8%B6%B31%2Fa1a2%2B1%2Fa2a3%2B%E2%80%A6%E2%80%A61%2Fan-1an%3D%28n-1%29%2Fa1an%2C%E6%B1%82%E8%AF%81%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%E6%BB%A1%E8%B6%B31%2F%28a1a2%29%2B1%2F%28a2a3%29%2B%E2%80%A6%E2%80%A61%2F%28an-1an%29%3D%28n-1%29%2Fa1an%2C%E6%B1%82%E8%AF%81%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.)
【急】已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列已知数列an满足1/(a1a2)+1/(a2a3)+……1/(an-1an)=(n-1)/a1an,求证为等差数列.
【急】已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列
已知数列an满足1/(a1a2)+1/(a2a3)+……1/(an-1an)=(n-1)/a1an,求证为等差数列.
【急】已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列已知数列an满足1/(a1a2)+1/(a2a3)+……1/(an-1an)=(n-1)/a1an,求证为等差数列.
n=3时
1/a1a2 + 1/a2a3= 2/a1a3
两边乘以 a1a2a3
得到 a3+a1=2a2 前三项满足等差数列
当n>=3时
1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an ①
1/a1a2+1/a2a3+……1/an-2an-1=(n-2)/a1an-1 ②
①-②--->(n-1)/a1an-(n-2)/a1an-1 = 1/an an-1
两边同时乘以 a1anan-1
得到: (n-1)an-1 - (n-2)an=a1 ③
同理 取n-1时
(n-2)an-2 - (n-3)an-1 =a1 ④
③-④---> (n-1)an-1 - (n-2)an = (n-2)an-2 - (n-3)an-1
(n-1+n-3)an-1 = (n-2)an +(n-2)an-2
2(n-2)an-1=(n-2)an+(n-2)an-2
n>=3
所以 2an-1 = an +an-2 符合等差数列定义
得证