Question

Let $a,x,b$ be in AP; $a,y,b$ be in GP and $a,z,b$ be in HP. If $x=y+2$ and $a=5z$ then

• A. $y^2=xz$
• B. $x>y>z$
• C. $a=9,b=1$
• D. $a=\frac{1}{4}$, $b=\frac{9}{4}$

Answer 1

Answer: (A), (B), (C)

$y^2=xz$
$x>y>z$
$a=9,b=1$

$a,x,b\to A.P\Rightarrow 2x=a+ba,y,b\to G.P\Rightarrow y^2=ab$
$a,z,b\to H.P.\Rightarrow z=\frac{2ab}{a+b}=\frac{{2y}^2}{2x}$
$\therefore xy=y^2$
Now for $a=9,b=1$
$x=5,y=3$
$\therefore 5=3+2\Rightarrow x=y+2$
And $z=\frac{2\times 9}{10}=\frac{9}{5}\Rightarrow a=5z$