若a+b+c=6,a^2+b^2+c^2=14,a^3+b^3+c^3=36,试求1/a+1/b+1/c的值.
问题描述：
若a+b+c=6,a^2+b^2+c^2=14,a^3+b^3+c^3=36,试求1/a+1/b+1/c的值.
答
(a+b+c)^2=a^2+b^2+c^2 +2ab+2bc+2ac=36
=>ab+bc+ca=(36-14)/2=11
(a+b+c)^3=a^3+b^3+c^3+3a^2b+3a^2c+3ab^2+3ac^2+3b^c+3bc^2+6abc
=3(a^2+b^2+c^2)(a+b+c)-2(a^3+b^3+c^3)+6abc=216
=>abc=6
所以1/a+1/b+1/c=ab+bc+ca/abc=11/6