Question

If $a,b,c$ are three unequal numbers such that $a,b,c$ are in $A.P.$ and $b-a,c-b,a-c$ are in $G.P,$ then $a:b:c$ is

• A. $1:2:3$
• B. $1:3:5$
• C. $2:3:4$
• D. $1:2:4$

Answer 1

The correct option is A $1:2:3$

In $A.P$

$\Rightarrow$ $(b-a)=(c-b)$ ----- ( 1 )

And in $G.P$

$\Rightarrow$ $(c-b{)}^2=a(b-a)$

$\Rightarrow$ $(b-a{)}^2=a(b-a)$ [ From ( 1 ) ]

$\Rightarrow$ $b-a=a$

$\Rightarrow$ $b=a+a$

$\Rightarrow$ $b=2a$

Now,

$\Rightarrow$ $(c-b{)}^2=a(b-a)$

$\Rightarrow$ $(c-b{)}^2=a(c-b)$ [ From ( 1 ) ]

$\Rightarrow$ $c-b=a$

$\Rightarrow$ $c=b+a$

Substituting value of $b$

$\Rightarrow$ $c=2a+a$

$\Rightarrow$ $c=3a$

$\Rightarrow$ $a:b:c:=a:2a:3a=1:2:3$