Question

If x , y , z are in A . P . ax , by , cz , in G . P . and a , b , c , in H . P . prove that $\frac{x}{z}+\frac{z}{x}=\frac{a}{c}+\frac{c}{a}$

Answer 1

By the given the condition , we have
2y = x + z
$b^2{y}^2=ax\dot{}cz$
b = 2ac /(a + c )
subsutiting from (1) (3) and (2) we get
$\frac{4a^2{c}^2}{(a+c{)}^2}\cdot \frac{(x+z{)}^2}{4}=axcz$

$\frac{(x+z{)}^2}{xz}\frac{(a+c{)}^2}{ac}$

$or\frac{x^2+z^2+2xz}{xz}=\frac{a^2+c^2+2ac}{ac}$

$or\frac{x}{z}+\frac{z}{x}+2=,\frac{a}{c}+\frac{c}{a}+2$

$or\frac{x}{z}+\frac{z}{x}=\frac{a}{c}+\frac{c}{a}$