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a1,a2,a3,a4是各项不为零的等差数列且公差d≠0,若将此数列删去某一项得到的数列(按原来的顺序)是等比数列,则
a1
d
的值为______.
人气:304 ℃ 时间:2020-04-03 21:34:57
解答
a2=a1+d  a3=a1+2d  a4=a1+3d
若a1、a2、a3成等比数列,则a22=a1•a3
(a1+d)2=a1(a1+2d)
a12+2a1d+d2=a12+2a1d
d2=0
d=0 与条件d≠0矛盾
若a1、a2、a4成等比数列,则a22=a1•a4
(a1+d)2=a1(a1+3d)
a12+2a1d+d2=a12+3a1d
d2=a1d
∵d≠0
∴d=a1
a1
d
=1
若a1、a3、a4成等比数列,则a32=a1•a4
(a1+2d)2=a1(a1+3d)
a12+4a1d+4d2=a12+3a1d
4d2=-a1d
∵d≠0
∴4d=-a1
a1
d
=-4
若a2、a3、a4成等比数列,则a32=a2•a4
(a1+2d)2=(a1+d)(a1+3d)
a12+4a1d+4d2=a12+4a1d+3d2
d2=0
d=0 与条件d≠0矛盾
综上所述:
a1
d
=1 或
a1
d
=-4
故答案为1或-4
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