If $$H1$$,$$H2$$,…,Hn are n harmonic means between a and $$b($$≠$$a)$$, then value of $$H1+aH1$$−$$a+Hn+bHn$$−bis equal to

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If $$H1$$,$$H2$$,…,Hn are n harmonic means between a and $$b($$≠$$a)$$, then value of $$H1+aH1$$−$$a+Hn+bHn$$−bis equal to

Infinity Learn · Accepted Answer

As a,$$H1$$,$$H2$$,…,Hnb are in H.P.    $$1a$$,$$1H1$$,$$1H2$$⋯$$1Hn$$,$$1b$$ are in A.P.Let dbe the common difference of this A.P., then            $$1b=1a+(n+1)dand$$  $$1Hn$$−$$1H1=(n$$−$$1)dNow$$, $$H1+aH1$$−$$a=1/a+1/H11/a$$−$$1/H1=1/a+1/H1$$−dand $$Hn+bHn$$−$$b=1/b+1/Hn1/b$$−$$1/Hn=1/b+1/Hnd$$∴    $$H1+aH1$$−$$a+Hn+bHn$$−$$b=1/a+1/H1$$−$$d+1/b+1/Hnd$$                                  $$=1d1b$$−$$1a+1Hn$$−$$1H1=2n$$

If $H_{1}, H_{2}, \dots, H_{n}$ are $n$ harmonic means between $a$ and $b (\neq a)$ , then value of $\frac{H_{1} + a}{H_{1} - a} + \frac{H_{n} + b}{H_{n} - b}$ is equal to

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If H1,H2,…,Hn are n harmonic means between a and b(≠a), then value of H1+aH1−a+Hn+bHn−bis equal to

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If $H_{1}, H_{2}, \dots, H_{n}$ are $n$ harmonic means between $a$ and $b (\neq a)$ , then value of $\frac{H_{1} + a}{H_{1} - a} + \frac{H_{n} + b}{H_{n} - b}$ is equal to