Question

If ‘a. b, c’ are in arithmetic progression ‘b, c, d’ are in geometric progression & ‘c, d, e’ are in harmonic progression, then ‘a, c, e’ are in

• A. Not particular order
• B. Arithmetic Progression
• C. Geometric Progression
• D. Harmonic Progression

Answer 1

The correct option is C

Geometric Progression

Explanation for correct option:

Finding the relation:

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

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

Since, ‘b, c, d’ are in geometric progression

$c^2=bd...\left(ii\right)$

Since, ‘c, d, e’ are in harmonic progression

$$\begin{gathered} \frac{2}{d}=\frac{1}{c}+\frac{1}{e} \\ \frac{2}{d}=\frac{c+e}{ce}...\left(iii\right) \end{gathered}$$

From equation $\left(i\right),\left(ii\right)\&\left(iii\right)$

$\begin{array}{rcl}\frac{2b}{c^2}& =& \frac{(c+e)}{ce}\\ \frac{a+c}{c}& =& \frac{(c+e)}{e}\\ \frac{(a+c)}{c}& =& \frac{(c+e)}{e}\\ ae+ce& =& c^2+ce\\ ae& =& c^2\end{array}$

So, ‘a, c, e’ are in geometric progression.

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