Question

Let a,x,b be in A.P; a,y, b be in G.P. and a,z,b be in H.P. If $x=y+2$ and $a=5z$ then:

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

Answer 1

The correct option is A $y^2=xz$

$a,x,b$-A.P
$x=\frac{a+b}{2}$
$2x=a+b\to 1$
$a,y,b$-G.P
$y^2=ab$
$a,z,b$-H.P.
$\frac{1}{a},\frac{1}{z},\frac{1}{b}$-A.P.
$\frac{2}{z}=\frac{1}{b}+\frac{1}{a}\to 2$

$\frac{2}{z}=\frac{a+b}{ab}$

From $1$ and $2$

$y^2=ab$
$2x=a+b$
$\frac{2}{z}=\frac{a+b}{ab}=\frac{2x}{yz}$
$y^2=xz$