Question

If ‘a, b, c’ are in arithmetic progression $b-a,c-b\&a$ are in geometric progression 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

$1:2:3$

Explanation for correct option.

Since, ‘a, b, c’ are in arithmetic progression

$\begin{array}{rcl}(b–a)& =& (c-b)\dots \left(i\right)\end{array}$

$b–a,c–b\&a$ are in geometric progression

So, $\begin{array}{rcl}{(c–b)}^2& =& (b-a)a...\left(ii\right)\end{array}$

From $\left(i\right)$ & $\left(ii\right)$

$\begin{array}{lcl}\Rightarrow {(c–b)}^2& =& (b-a)a\\ \Rightarrow {(b–a)}^2& =& (b-a)a\\ \Rightarrow (b–a)& =& a\\ \Rightarrow 2a& =& b\end{array}$

Put the value of ‘b’ in equation $\left(i\right)$

Therefore,

$\begin{array}{rcl}2a-a& =& c-2a\\ c& =& 3a\end{array}$

Hence, $a:b:c=1:2:3$

Hence, the correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{.}$