First slide

Harmonic Progression

Question

$If H_{P} H_{2}, \dots . H_{20} harmonic means between 2 and 3, then \frac{H_{1} + 2}{H_{1} - 2} + \frac{H_{20} + 3}{H_{20} - 3}$

Moderate

A

20
B

21
C

40
D

38

Solution

$\begin{array}{l} 2, H_{1}, H_{2} \dots \dots . H_{20}, 3 are in H. P \\ \frac{H_{1} + 2}{H_{1} - 2} + \frac{H_{20} + 3}{H_{20} - 3} = \frac{\frac{1}{2} + \frac{1}{H_{1}}}{\frac{1}{2} - \frac{1}{H_{1}}} + \frac{\frac{1}{3} + \frac{1}{H_{20}}}{\frac{1}{3} - \frac{1}{H_{20}}} \\ = \frac{\frac{1}{2} + \frac{1}{2} + d}{\frac{1}{2} - d - \frac{1}{2}} + \frac{\frac{1}{3} + \frac{1}{3} - d}{\frac{1}{3} + d - \frac{1}{3}} \end{array}$
$= \frac{1 + d}{- d} + \frac{\frac{2}{3} - d}{d} = \frac{\frac{2}{3} - 1}{d} - 2 = 2 \times 21 - 2 = 40$

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