设实数a,b,c成等差数列，非实数x,y分别为a与b,b与c的等差中项，求证：（a/x）+c/y=
\u8bbe\u5b9e\u6570a,b,c\u6210\u7b49\u5dee\u6570\u5217,\u975e\u96f6\u5b9e\u6570x,y\u5206\u522b\u4e3aa\u4e0eb,b\u4e0ec\u7684\u7b49\u5dee\u4e2d\u9879,\u6c42\u8bc1a/x+c/y=2\u9898\u76ee\u6709\u8bef
\u5982\u679c\u539f\u9898\u6210\u7acb\uff0c\u5219a,x,b,y,c\u6210\u7b49\u5dee\u6570\u5217
\u5047\u8bbea=1,x=2,b=3,y=4,c=5\uff0c\u8be5\u6761\u4ef6\u6ee1\u8db3\u9898\u610f
a/x+c/y=1/2+5/4=7/4\uff0c\u663e\u7136\u4e0d\u7b49\u4e8e2
\u5982\u679c\u539f\u9898\u6539\u4e3aa,b,c\u6210\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u6c42\u8bc1
2x=a+b 2y=b+c
a/x+c/y
=2a/(a+b)+2c/(b+c)
=2/(1+b/a)+2/(b/c+1)
\u2235a,b,c\u6210\u7b49\u6bd4
\u2234\u8bbe\u516c\u6bd4\u4e3aq\uff0cq=b/a=c/b
a/x+c/y=2/(1+b/a)+2/(b/c+1)
=2/(1+q)+2/(1/q+1)
=2/(1+q)+2q/(1+q)
=(2+2q)/(1+q)
=2
\u8bc1\u5f97
a\u3001b\u3001c\u6210\u7b49\u5dee\u6570\u5217
a+c=2b
a\u3001c\u3001b\u6210\u7b49\u6bd4\u6570\u5217
c^2=ab
a/c=c/b
\u8bbe\uff1a
a/c=c/b=k
a=ck
c=bk
a=bk^2
\u4ee3\u5165a+c=2b
bk^2+bk=2b
k^2+k-2=0
(k+2)(k-1)=0
k1=1
k2=-2
k=1\u65f6\uff0ca=b=c \u820d
k=-2\u65f6 a=4b c=-2b
a:b:c=4b:b:(-2b)=4:1:(-2)
证明：因为a，b，c成等比数列
所以 b2=ac①
又x，y分别为a与b，b与c的等差中项
所以 2x=a+b，2y=b+c②
要证a/x+c/y=2
只要证 ay+cx=2xy
只要证 2ay+2cx=4xy
由①②得 2ay+2cx=a（b+c）+c（a+b）=ab+2ac+bc；
而4xy=（a+b）（b+c）=ab+b2+ac+bc=ab+2ac+bc成立．
所以命题得证．
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