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Relation between AM,GM,HM for 2 Numbers

If n harmonic...

Question

If n harmonic means are inserted between 1 and r, then show that $\frac{1 s t m e a n}{n t h m e a n} = \frac{n + r}{n r + + 1}$

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Solution

We have calculated 1st harmonic mean in (2) of last question
$y_{1} = \frac{(n + 1) a b}{n b + a}$
$∴ y_{n} = n t h H . M . = \frac{(n + 1) a b}{n b + a}$
$∴ \frac{1 s t m e a n}{n t h m e a n} = \frac{y_{1}}{y_{n}} = \frac{b + n a}{n b + a}$
$= \frac{r + n .1}{n r + 1} = \frac{n + r}{n r + 1}$
$[∴$ a = 1, b = r]

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