Question

已知x、y、z、∈R,3x,4y,5z成等比数列.1/x,1/y,1/z成等差数列,则z/x+x/z=＿＿

Answer 1

(4*y)^2=3*x*5*z ,即16*y^2=15*x*z.

2*1/y=1/x+1/z=(x+z)/(x*z),即2/y=(x+z)/(x*z)

z/x+x/z

=(x^2+z^2)/(x*z)

=[(x+z)^2-2*x*z]/(x*z)

=(x+z)^2/(x*z)-2

=(x+z)^2*（x*z)/(x*z)^2-2

=(2/y)^2*(16*y^2/15)-2

=64/15-2

=34/15