已知Sn是等比数列an中的前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中的前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中的前n项和,S3.S9.S6成等差数列,求证a2.a8.a5成等差数列.](/uploads/image/z/5601349-37-9.jpg?t=%E5%B7%B2%E7%9F%A5Sn%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97an%E4%B8%AD%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2CS3.S9.S6%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%B1%82%E8%AF%81a2.a8.a5%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.)
因为S3.S9.S6成等差数列
2S9=S3+S6
2a1(1-q^8)/(1-q)=a1(1-q^2)/(1-q)+a1(1-q^5)/(1-q)
2(1-q^8)=2-q^2-q^5
2q^8=q^2+q^5
2q^7=q+q^4
2a1q^7=a1q+a1q^4
2a8=a2+a5
所以a2.a8.a5成等差数列
2S9=S3+S6
2a1(1-q^8)/(1-q)=a1(1-q^2)/(1-q)+a1(1-q^5)/(1-q)
2(1-q^8)=2-q^2-q^5
2q^8=q^2+q^5
2q^7=q+q^4
2a1q^7=a1q+a1q^4
2a8=a2+a5
所以a2.a8.a5成等差数列
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