设a、b、c满足1/a+1/b+1/c=1/a+b+c,求证:当n为奇数时,1/an+bn+cn=1/an+1/bn+1/cn.
问题描述:
设a、b、c满足
+1 a
+1 b
=1 c
,求证:当n为奇数时,1 a+b+c
=1
an+bn+cn
+1 an
+1 bn
. 1 cn
答
证明:由已知,得1a+1b=1a+b+c-1c,通分,得a+bab=c−a−b−cc(a+b+c),去分母、移项,得c(a+b)(a+b+c)+ab(a+b)=0,(a+b)(ac+bc+c2+ab)=0(a+b)(b+c)(a+c)=0即a=-b,b=-c,c=-a至少有一个成立,故当n...