已知Sn是等比数列〔Aa〕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〔Aa〕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〔Aa〕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/8509234-58-4.jpg?t=%E5%B7%B2%E7%9F%A5Sn%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E3%80%94Aa%E3%80%95%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%EF%BC%8ES3.S9.S6%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%EF%BC%8E%E6%B1%82%E8%AF%81A2.A8.A5%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97)
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.
证明:由已知设An=A1q^(n-1),q为公比且不为0.
则q不为0也不为1时,Sn=[(q^n-1)/(q-1)]A1;
当q=1时,Sn=nA1;
∵2S9=S3+S6,
∴2[(q^9-1)/(q-1)]A1=[(q^3-1)/(q-1)]A1+[(q^6-1)/(q-1)]A1,
∴2q^9=q³+q^6,∴q³(2q³+1)(q³-1)=0,
∴q³=1或-1/2,∴q=1或(-1/2)^(1/3),
当q=1时,An=A1,则2A8-(A8+A5)=2A1-(A1+A1)=0,得证;
当q=(-1/2)^(1/3)时,An=A1[(-1/2)^(1/3)]^(n-1);
2A8-(A8+A5)
=2×A1[(-1/2)^(1/3)]^(8-1)
-{[A1[(-1/2)^(1/3)]^(2-1)]+[A1[(-1/2)^(1/3)]^(5-1)]}
=(A1/2)(-1/2)^(1/3)-[A1(-1/2)^(1/3)+(-A1/2)(-1/2)^(1/3)]
=(A1/2)(-1/2)^(1/3)-(A1/2)(-1/2)^(1/3)
=0,得证.
求证:A2、A8、A5成等差数列.
证明:由已知设An=A1q^(n-1),q为公比且不为0.
则q不为0也不为1时,Sn=[(q^n-1)/(q-1)]A1;
当q=1时,Sn=nA1;
∵2S9=S3+S6,
∴2[(q^9-1)/(q-1)]A1=[(q^3-1)/(q-1)]A1+[(q^6-1)/(q-1)]A1,
∴2q^9=q³+q^6,∴q³(2q³+1)(q³-1)=0,
∴q³=1或-1/2,∴q=1或(-1/2)^(1/3),
当q=1时,An=A1,则2A8-(A8+A5)=2A1-(A1+A1)=0,得证;
当q=(-1/2)^(1/3)时,An=A1[(-1/2)^(1/3)]^(n-1);
2A8-(A8+A5)
=2×A1[(-1/2)^(1/3)]^(8-1)
-{[A1[(-1/2)^(1/3)]^(2-1)]+[A1[(-1/2)^(1/3)]^(5-1)]}
=(A1/2)(-1/2)^(1/3)-[A1(-1/2)^(1/3)+(-A1/2)(-1/2)^(1/3)]
=(A1/2)(-1/2)^(1/3)-(A1/2)(-1/2)^(1/3)
=0,得证.
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