数列{An}是等比数列Sn是前n项和,A1,A7,A4成等差数列求证2S3,S6,S12-S6成等比数列
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数列{An}是等比数列Sn是前n项和,A1,A7,A4成等差数列求证2S3,S6,S12-S6成等比数列
![数列{An}是等比数列Sn是前n项和,A1,A7,A4成等差数列求证2S3,S6,S12-S6成等比数列](/uploads/image/z/16071023-47-3.jpg?t=%E6%95%B0%E5%88%97%7BAn%7D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97Sn%E6%98%AF%E5%89%8Dn%E9%A1%B9%E5%92%8C%2CA1%2CA7%2CA4%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E6%B1%82%E8%AF%812S3%2CS6%2CS12-S6%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97)
a7=a1q^6
a4=a1q^3
因为a1,a7,a4成等差数列
所以a7=(a1+a4)/2
a1q^6=(a1+a1q^3)/2
2q^6-q^3-1=0
(2q^3+1)(q^3-1)=0
q^3=1,或q^3=-1/2
因为
2S3=2a1(1-q^3)/(1-q)
S6=a1(1-q^6)/(1-q)
S12-S6=a1(1-q^12)/(1-q)-a1(1-q^6)/(1-q)
S6/2S3=(1+q^3)/2
(S12-S6)/S6=(1+q^6)-1
当q^3=1时,S6/2S3=(1+q^3)/2=1,(S12-S6)/S6=(1+q^6)-1
=1
当q^3=-1/2时,S6/2S3=(1+q^3)/2=1/4,(S12-S6)/S6=(1+q^6)-1
=1/4
所以2S3,S6,S12-S6成等比数列
a4=a1q^3
因为a1,a7,a4成等差数列
所以a7=(a1+a4)/2
a1q^6=(a1+a1q^3)/2
2q^6-q^3-1=0
(2q^3+1)(q^3-1)=0
q^3=1,或q^3=-1/2
因为
2S3=2a1(1-q^3)/(1-q)
S6=a1(1-q^6)/(1-q)
S12-S6=a1(1-q^12)/(1-q)-a1(1-q^6)/(1-q)
S6/2S3=(1+q^3)/2
(S12-S6)/S6=(1+q^6)-1
当q^3=1时,S6/2S3=(1+q^3)/2=1,(S12-S6)/S6=(1+q^6)-1
=1
当q^3=-1/2时,S6/2S3=(1+q^3)/2=1/4,(S12-S6)/S6=(1+q^6)-1
=1/4
所以2S3,S6,S12-S6成等比数列
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