已知Sn是等比数列{an}的前n项和,S3,S9,S6成等差数列,求证a2,a8,a5成等差数列.
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已知Sn是等比数列{an}的前n项和,S3,S9,S6成等差数列,求证a2,a8,a5成等差数列.
![已知Sn是等比数列{an}的前n项和,S3,S9,S6成等差数列,求证a2,a8,a5成等差数列.](/uploads/image/z/685934-62-4.jpg?t=%E5%B7%B2%E7%9F%A5Sn%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2CS3%2CS9%2CS6%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%B1%82%E8%AF%81a2%2Ca8%2Ca5%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.)
首项a
2S9=2a(q^9-1)/(q-1)
S3+S6=a(a^3-1)/(q-1)+a(a^6-1)/(q-1)
2S9=S3+S6
显然a不等于0
2(q^9-1)=a^3-1+q^6-1
2q^9=q^3+q^6
2q^7=q+q^4
2aq^7=aq+aq^4
2a8=a2+a5
所以a2,a8,a5成等差数列
2S9=2a(q^9-1)/(q-1)
S3+S6=a(a^3-1)/(q-1)+a(a^6-1)/(q-1)
2S9=S3+S6
显然a不等于0
2(q^9-1)=a^3-1+q^6-1
2q^9=q^3+q^6
2q^7=q+q^4
2aq^7=aq+aq^4
2a8=a2+a5
所以a2,a8,a5成等差数列
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