Question

If $a,b,c$ are in A.P. and $a,c-b,b-a$ are in G.P. $(a\ne b\ne c),$ then $a:b:c$ is

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

Answer 1

The correct option is C 1 : 2 : 3

Here, $b=\frac{1}{2}$(a+c) and $(c-b{)}^2=a(b-a)$

$\Rightarrow {[c-\frac{1}{2}(a+c\left)\right]}^2=a\left[\frac{1}{2}\right(a+c)-a]$

$\Rightarrow c^2+3a^2=4ac$
$\Rightarrow c^2+a^2-2ac+2a^2=2ac$
$\Rightarrow (c-a{)}^2=2a(c-a)$
$\Rightarrow c-a=2a$
Since $a\ne c,\Rightarrow c=3a$, so, $b=2a$

∴ $\frac{a}{1}=\frac{b}{2}=\frac{c}{3}$ i.e., $a:b:c=1:2:3.$