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

The correct option is A

$1:2:3$

Step 1: Use A.P relation between$a,b\&c$.

We have given$a,bandc$ are three unequal numbers and $a,bandc$are in A.P and $b-a,c-banda$are in G.P

we have to find out$a:b:c$.

as $a,bandc$are in A.P,

$\therefore b-a=c-b.....\left(1\right)$

Step 2: Use G.P relation.

as $b-a,c-banda$are in G.P

$\begin{array}{rcl}\therefore {(c-b)}^2& =& (b-a)a\\ {(b-a)}^2& =& (b-a)a\\ (b-a)& =& a\\ b& =& a+a\\ b& =& 2a....\left(2\right)\end{array}$ from$\left(1\right)$

Step 3: Find$c$ in the form of$a$.

From$\left(1\right)\&\left(2\right)$,

$\begin{array}{rcl}2a-a& =& c-2a\\ a+2a& =& c\\ 3a& =& c....\left(3\right)\end{array}$

So, $\begin{array}{rcl}a:b:c& =& a:2a:3a\end{array}$

Hence, the option$\left(A\right)1:2:3$ is correct.