设a,b,c是实数,3a,4b,5c成等比数列,且1/a,1/b,1/c成等比数列,求a/c+c/a

题目

设a,b,c是实数,3a,4b,5c成等比数列,且1/a,1/b,1/c成等比数列,求a/c+c/a

答案

3a,4b,5c成等比数列;
(4b)^2=3a*5c;
ac=16/15b^2;
1/a,1/b,1/c成等差数列;
1/a+1/c=2/b;
a+c/ac=2/b;
a+c/(16/15)b^2=2/b;;
a+c=32/15b;
a,c为方程x^2-32/15bx+16/15b^2=0;的两个跟；
a=4/3b;c=4/5b;或a=4/5b;c=4/3b;
所以a/c+c/a=[4/3b]/[4/5b]+[4/5b]/[4/3b]/=5/3+3/5=34/15;
或a/c+c/b=[4/5b]/[4/3b]+[4/3b]/b=3/5+4/5=34/15 '
a/c+c/a的值为34/15 ;