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(A) The first $n$ terms of the geometric progression are $a, ar, ar^2, \dots, ar^{n-1}$.The arithmetic mean $A$ is given by:$A = \frac{a + ar + ar^2 +

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$a^2 r^{n-1}$

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(A) The first $n$ terms of the geometric progression are $a, ar, ar^2, \dots, ar^{n-1}$.The arithmetic mean $A$ is given by:$A = \frac{a + ar + ar^2 + \dots + ar^{n-1}}{n} = \frac{a(r^n - 1)}{n(r - 1)}$.The harmonic mean $H$ is given by:$H = \frac{n}{\frac{1}{a} + \frac{1}{ar} + \dots + \frac{1}{ar^{n-1}}} = \frac{n}{\frac{1}{a} \left( \frac{1 - (1/r)^n}{1 - 1/r} \right)} = \frac{n}{\frac{1}{a} \left( \frac{r^n - 1}{r^n} \cdot \frac{r}{r - 1} \right)} = \frac{n a r^{n-1} (r - 1)}{r^n - 1}$.Multiplying $A$ and $H$:$A \cdot H = \left( \frac{a(r^n - 1)}{n(r - 1)} \right) \cdot \left( \frac{n a r^{n-1} (r - 1)}{r^n - 1} \right) = a^2 r^{n-1}$.

Consider a geometric progression with first term $a$ and common ratio $r$. If $A$ and $H$ are the arithmetic mean and harmonic mean of the first $n$ terms of the geometric progression respectively,then $A \cdot H = \dots$

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