Question

If $a,x,b$ are in AP, $a,y,b$ are in GP and $a,z,b$ are in HP such that $x=9z$ and $a>0,b>0$ then

• A. $y=3z$
• B. $x=3y$
• C. $2y=x+z$
• D. none of these

Answer 1

Answer: (A), (B)

$y=3z$
$x=3y$

a, x, b are in AP
$\Rightarrow x=\frac{a+b}{2}$ _____ (1)
a, y, b are in GP
$\Rightarrow y^2=ab$ _____ (2)
a, z, b are in HP
$\Rightarrow 2z=\frac{2ab}{a+b}$
$\Rightarrow z=\frac{ab}{x}$ from (1)
Also $x=9z$
$\Rightarrow z=\frac{ab}{9z}$
$\Rightarrow 9z^2=y^2$ from (2)
$\therefore 3z=y$
$\therefore x=3y$
Hence, options A and B are correct.