First slide

Harmonic Progression

Question

If $a, x, b$ are in H.P. and $a, y, z, b$ are in G.P., then the value of $\frac{y z}{x (y^{3} + z^{3})}$ is

Moderate

A

$a b$
B

$\frac{1}{2 a b}$
C

$\frac{1}{2} a b$
D

$2 a b$

Solution

$x = \frac{2 a b}{a + h}$
Let common ratio of G.P. $a, y, z, b$
be $r,$ then $r^{3} = b / a$ and
$\begin{array}{l} y = a r, z = b / r \\ ∴ y z = a b \end{array}$
and $y^{3} + z^{3} = a^{3} r^{3} + \frac{b^{3}}{r^{3}}$
$\begin{array}{l} = a^{3} (\frac{b}{a}) + b^{3} (\frac{a}{b}) \\ = a b (a + b) \\ \Rightarrow x (y^{3} + z^{3}) = \frac{2 a b}{a + b} (a b) (a + b) = 2 a^{2} b^{2} \\ ∴ \frac{y z}{x (y^{3} + z^{3})} = \frac{a b}{2 (a b)^{2}} = \frac{1}{2 a b} \end{array}$

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