Let there be a G.P. whose first term is ' $a$ ' and common ratio is ' $r$ '. If $A$ and $H$ are the

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Let there be a G.P. whose first term is ' $a$ ' and common ratio is ' $r$ '. If $A$ and $H$ are the arithmetic mean and the harmonic mean respectively for the first ' $n$ ' terms of the G.P. Then AH is equal to

Accepted Answer

Sol. [Key Concept: For the first two G.P. terms, $A = \frac{a(1 + r)}{2}$ and $H = \frac{2ar}{1 + r}$; combine as required.]$$A = \frac{a\left( 1 + r + r^{2} + \ldots r^{n - 1} ight)}{n}$$$$\begin{matrix} H & \ = \frac{n}{\frac{1}{a}\left( 1 + \frac{1}{r} + \frac{1}{r^{2}} + \ldots\frac{1}{r^{n - 1}} ight)} = \frac{{nar}^{n - 1}}{1 + r + r^{2} + \ldots + r^{n - 1}} \ AH & \ = a^{2}r^{n - 1} \end{matrix}$$

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