If H1,H2,H3,………Hn be n harmonic means between a and b thenH1+aH1−a+Hn+bHn−b=

A
A
B
B
C
N
D
2n

Solution:

$a, H_{1}, H_{2}, H_{3}, \dots \dots . H_{n}, b areinH . P \frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{H_{3}}, \dots \dots . . \frac{1}{H_{n}}, \frac{1}{b} areA . P$
$\begin{array}{ll} \frac{1}{H_{1}} = \frac{1}{a} + d, \frac{1}{H_{n}} = \frac{1}{b} - d \frac{H_{1} + a}{H_{1} - a} + \frac{H_{n} + b}{H_{n} - b} = \frac{\frac{1}{a} + \frac{1}{H_{1}}}{\frac{1}{a} - \frac{1}{H_{1}}} + \frac{\frac{1}{b} + \frac{1}{H_{n}}}{\frac{1}{b} - \frac{1}{H_{n}}} \\ = \frac{\frac{1}{a} + \frac{1}{a} + d}{\frac{1}{a} - \frac{1}{a} - d} + \frac{\frac{1}{b} + \frac{1}{b} - d}{\frac{1}{b} - \frac{1}{b} + d} = \frac{\frac{2}{a} + d}{- d} + \frac{\frac{2}{b} - d}{d} = \frac{\frac{2}{b} - d - \frac{2}{a} - d}{d} \end{array}$
$\begin{array}{l} = \frac{\frac{2}{b} - \frac{2}{a} - 2 d}{d} = \frac{\frac{2}{b} - \frac{2}{a}}{d} - 2 = \frac{\frac{2}{a} + 2 (n + 1) d - \frac{2}{a}}{d} - 2 (∴ \frac{1}{b} = \frac{1}{a} + (n + 2 - 1) d = \frac{1}{a} + (n + 1) d) \\ = \frac{2 d (n + 1)}{d} - 2 = 2 n + 2 - 2 = 2 n \end{array}$

If H₁,H₂,H₃,………Hn be n harmonic means between a and b then $\frac{H_{1} + a}{H_{1} - a} + \frac{H_{n} + b}{H_{n} - b} =$

Solution:

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