Question

If $a,b,c$ are in Arithmetic Progression and geometric means of $ac\&ab,ab\&bc,ca\&cbared,e,f$, respectively then $d^2,e^2,f^2$ are in

• A. Arithmetic Progression
• B. Geometric Progression
• C. Harmonic Progression
• D. None of these

Answer 1

The correct option is A

Arithmetic Progression

Explanation for correct option.

Step1. Expressing the given data:

Geometric Mean of $ac\&ab=d$

So,

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

Geometric mean of $ab\&bc=e$

So,$e^2=a{b}^{2}c$

Geometric mean of $ca\&cb=f$

So,$f^2=ab{c}^2$

Step2. Relation between ${\mathit{d}}^{\mathbf{2}}\mathbf{,}\mathbf{}{\mathit{e}}^{\mathbf{2}}\mathbf{,}\mathbf{}{\mathit{f}}^{\mathbf{2}}$

Since $a,b,c$are in Arithmetic Progression

Therefore, $b=\frac{a+c}{2}$

Now,

$\begin{array}{rcl}{d}^2+f^2& =& a^{2}bc+ab{c}^2\\ & \Rightarrow & d^2+f^2=acb(a+c)\\ & \Rightarrow & d^2+f^2=\frac{(a+c)}{2}2abc\\ & \Rightarrow & d^2+f^2=b\times 2abc\\ & \Rightarrow & d^2+f^2=2ac{b}^2\\ & \Rightarrow & d^2+f^2=2e^2\end{array}$

So they are in arithmetic progression

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