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Harmonic Mean

If 0<r<1 an...

Question

If $0 < r < 1$ and $n$ is a $+ i v e$ integer then prove that $(2 n + 1) r^{n} (1 - r) < 1 - r^{2 n + 1}$ .

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Solution

We have to prove that $\frac{1 - r^{2 n + 1}}{1 - r} > (2 n + 1) r^{2}$
L.H.S. = Sum of a $G . P .$ of $2 n + 1$ terms whose first terms is $1$ , and common ratio $r$ .
L.H.S. $= 1 + r + r^{2} + . . . + r^{2 n}$
$= (1 + r^{2 n}) + (r + r^{2 n - 1}) + (r^{2} + r^{2 n - 2}) + . . . n$ pairs $+ r^{n}$
Apply $A . M . > G . M .$ on each pair
$∴ L . H . S . > ({{2 r}^{n} + 2 r}^{n} + . . . + {2 r}^{n} n t e r m s) + r^{n}$
$= 2 n . r^{n} + r^{n} = (2 n + 1) r^{n}$

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