Question

If $a,b,c$ are in $A.P$ and $a,b,d$in $G.P.$, then $a,a-b,d-c$ will be in

• A. $A.P$
• B. $G.P$
• C. $H.P$
• D. None of these

Answer 1

The correct option is B

$G.P$

Step 1. Finding the progression:

Given that, $a,b,c$ are in $AP$ and $a,b,d$in $GP$

Then, $2b=a+c...\left(i\right)$

$\because a,b,d$in $GP$

$\therefore$$b^2=ad\left(ii\right)$

Step 2. Multiply equation $\left(i\right)$ by $\text{'}a\text{'}$

$\begin{array}{rcl}2ab& =& a^2+ac\end{array}$

$\Rightarrow$ $\begin{array}{rcl}-ac& =& a^2-2ab\end{array}$

Step 3. Add $b^2$on both side,

$\begin{array}{rcl}{b}^2-ac& =& a^2-2ab+b^2\end{array}$

$\Rightarrow$ $\begin{array}{rcl}ad-ac& =& {(a-b)}^2\end{array}$ ; $\begin{array}{rcl}[\therefore b^2& =& ad]\end{array}$

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

$\mathbf{\therefore }\mathit{a}\mathbf{,}\mathbf{}\mathbf{(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{(}\mathit{d}\mathbf{-}\mathit{c}\mathbf{)}$ are in $\mathit{G}\mathit{P}$

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