【高中数学证明题一道】设a1>a2>…>an>an+1,求证1/(a1-a2)+1/(a2-a3)+…+1/(an-an
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【高中数学证明题一道】设a1>a2>…>an>an+1,求证1/(a1-a2)+1/(a2-a3)+…+1/(an-an+1)+1/(an+1-a1)>0.
设a1>a2>…>an>an+1,求证1/(a1-a2)+1/(a2-a3)+…+1/(an-an+1)+1/(an+1-a1)>0.
最好能用上柯西不等式或均值不等式。
设a1>a2>…>an>an+1,求证1/(a1-a2)+1/(a2-a3)+…+1/(an-an+1)+1/(an+1-a1)>0.
最好能用上柯西不等式或均值不等式。
![【高中数学证明题一道】设a1>a2>…>an>an+1,求证1/(a1-a2)+1/(a2-a3)+…+1/(an-an](/uploads/image/z/1286355-3-5.jpg?t=%E3%80%90%E9%AB%98%E4%B8%AD%E6%95%B0%E5%AD%A6%E8%AF%81%E6%98%8E%E9%A2%98%E4%B8%80%E9%81%93%E3%80%91%E8%AE%BEa1%3Ea2%3E%E2%80%A6%3Ean%3Ean%2B1%2C%E6%B1%82%E8%AF%811%2F%28a1-a2%29%2B1%2F%28a2-a3%29%2B%E2%80%A6%2B1%2F%28an-an)
因为 1/(an+1-a1)+1/(a1-an+1)=0
所以 只需证明 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)>1/(a1-an+1)
因为a1>a2>a3...>an>an+1
所以 a1>an
a1-an+1>an-an+1>0
1/(an-an+1)>1/(a1-an+1)
所以 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)>1/(an-an+1)>1/(a1-an+1)
所以 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)+1/(an+1-a1)>0
所以 只需证明 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)>1/(a1-an+1)
因为a1>a2>a3...>an>an+1
所以 a1>an
a1-an+1>an-an+1>0
1/(an-an+1)>1/(a1-an+1)
所以 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)>1/(an-an+1)>1/(a1-an+1)
所以 1/(a1-a2)+1/(a2-a3)+...+1/(an-an+1)+1/(an+1-a1)>0
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