First slide

Harmonic Progression

Question

$a, b, c, d \in R^{+}$ such that a, b and c are in A.P. and b, c and d are in H.P., then

Moderate

A

ab=cd
B

ac=bd
C

bc=ad
D

none of these

Solution

$\begin{array}{l} 2 b = a + c, c = \frac{2 bd}{b + d} \\ \Rightarrow 2 bd = c (b + d) \\ \Rightarrow (a + c) d = c (b + d) [as 2 b = a + c] \\ \Rightarrow ad + cd = bc + cd \\ \Rightarrow bc = ad \end{array}$

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