设Sn是等差数列｛an｝的前N项和,诺S3/S6=1/3则S6/S12=多少

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设Sn是等差数列｛an｝的前N项和,诺S3/S6=1/3则S6/S12=多少
设Sn是等差数列｛an｝的前N项和,诺S3/S6=1/3则S6/S12=多少

设Sn是等差数列｛an｝的前N项和,诺S3/S6=1/3则S6/S12=多少
S3=A1+A2+A3=3A1+3d
S6=A1+A2+.+A6=6A1+15d
(3A1+3d)/(6A1+15d) = 1/3
3(A1+d)=(2A1+5d
A1=2d
A2=3d A3=4d
S6/S12=(2d+3d+4d+5d+6d+7d)/(2d+3d+.+13d) =27/88

S3/S6=1/3
S6=3S3
(a1+a6)*6/2=3(a1+a3)*3/2
a1+a6=3(a1+a3)/2
所以2a1+2a6=3a1+3a3
2a6=a1+3a3
2(a1+5d)=a1+3(a1+2d)
a1=2d
S6/S12
=[(a1+a6)*6/2]/[(a1+a12)*12/2]
=(a1+a1+5d)/2(a1+a1+11d)
=5d/(2*15d)
=1/6

S3=3a+3d
S6=6a+15d
S12=12a+66d
S3/S6=1/3
则（3a+3d）/（6a+15d）=1/3
解得a=2d
所以S6/S12=3/10
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等差数列S3,S6-S3,S9-S6,S12-S9...

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S3=3a+3d
S6=6a+15d
S12=12a+66d
S3/S6=1/3
则（3a+3d）/（6a+15d）=1/3
解得a=2d
所以S6/S12=3/10
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等差数列S3,S6-S3,S9-S6,S12-S9也成等差数列
S3/S6=1/3,S6=3S3,S6-S3=2S3
S9-S6=3S3,S9=6S3
S12-S9=4S3,S12=10S3
所以S6/S12=3/10

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