First slide

Geometric progression

Question

Let $a, b, c, d$ and $e$ be distinct positive number. If $a, b, c$ and $\frac{1}{c}, \frac{1}{d}, \frac{1}{e}$ both are in A.P. and $b, c, d$ are in G.P. then

Moderate

A

$a, b, c$ are in G.P.
B

$a, b, c$ are in A.P
C

$a, c, e$ are in A.P.
D

$a, c, e$ are in G.P

Solution

$2 b = a + c, \frac{2}{d} = \frac{1}{c} + \frac{1}{e}, c^{2} = b d$
Now, $\frac{2 c e}{c + e} = d = \frac{c^{2}}{b}$
$\begin{array}{l} \Rightarrow \frac{e}{c + e} = \frac{c}{2 b} = \frac{c}{a + c} \\ \Rightarrow a e + c e = c^{2} + c e \end{array}$
$\Rightarrow c^{2} = a e \Rightarrow a, c, e$ are in G.P.

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