设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12

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设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12设Sn是等差数列{an}的前n项和,若S3/S6=1/3

设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12
设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12

设Sn是等差数列{an}的前n项和,若S3/S6=1/3,求S6/S12
首先你要知道等差数列的顺次n项和性质
即Sn,S2n-Sn,S3n-S2n成公差为n²d的等差数列
则S6-S3=S3+9d
由S3/S6得S3=9d
则S6=27d d=S6/27
S9-S6=S6-S3+9d得S9=2S6
S12-S9=S9-S6+9d得S12=3S6+9d
得S6/S12=S6/(3S6+S6/3)=3/10

s6/s12=5/22

3S3=S6. S3,S6-S3,S9-S6,S12-S9仍为等差数列，所以S9=S6-S3+2(S6-S3)=6S3因为S3+S12-S9=(S6-S3)+(S9-S6),所以S12=2S9-2S3= 10S3所以S6/S12=3/10

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在等差数列中有Sn,S2n-Sn,S3n-S2n……成等差数列
S6=3S3
S6-S3=2S3
S3,S6-S3,S9-S6,S12-S9成等差数列（证明见最后）
S9-S6=3S3 S12-S9=4S3
S9=S6+3S3=6S3
S12=S9+4S3=10S3
S6/S12=3S3/10S3=3/10
证明S3,S6-S3,S9-S6,S12-S9成等差数列
S3=A1+A2+A3
S6-S3=A4+A5+A6=(A1+3d)+(A2+3d)+(A3+3d)=A1+A2+A3+9d=S3+9d
S9-S6=A7+A8+A9=A1+A2+A3+18d=S3+18d
S12-S9=A10+A11+A12=A1+A2+A3+27d=S3+27d
S3,S6-S3,S9-S6,S12-S9成等差数列

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只能估计d=2

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