设Sn是等差数列前n项和,已知1/3S3,1/4S4的等比中项为1/5S5,1/3S3,1/4S4的等差中项为1,求等差数列an的通项

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设Sn是等差数列前n项和,已知1/3S3,1/4S4的等比中项为1/5S5,1/3S3,1/4S4的等差中项为1,求等差数列an的通项
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设Sn是等差数列前n项和,已知1/3S3,1/4S4的等比中项为1/5S5,1/3S3,1/4S4的等差中项为1,求等差数列an的通项
设Sn是等差数列前n项和,已知1/3S3,1/4S4的等比中项为1/5S5,1/3S3,1/4S4的等差中项为1,求等差数列
an的通项

设Sn是等差数列前n项和,已知1/3S3,1/4S4的等比中项为1/5S5,1/3S3,1/4S4的等差中项为1,求等差数列an的通项
∵a1+a3=2a2,∴S3=a1+a2+a3=3a2,∴(1/3)S3=a2
同理：a1+a4=a2+a3,∴S4=a1+a2+a3+a4=2(a2+a3),∴(1/4)S4=(a2+a3)/2
a1+a5=a2+a4=2a3,∴S5=a1+a2+a3+a4+a5=5a3,∴(1/5)S5=a3
由题意得
(1/3)S3×(1/4)S4=[(1/5)S5]²
(1/3)S3+(1/4)S4=2×1
即：
a2(a2+a3)/2=a3²
a2 + (a2+a3)/2=2
解得
a2=8/5,a3=-4/5
或a2=1,a3=1
①当a2=8/5,a3=-4/5时,
d= -12/5,a1=4
∴an=4-(12/5)(n-1)=(32-12n)/5
②当a2=1,a3=1时,
d=0,a1=1
∴an=1

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