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Question
Write the product of n geometric means between two numbers a and b.
Solution
\[\text{ Let G_1 , G_2 , . . . , G_n be n geometric means between two quantities a and b } . \]
\[\text{ Thus, a, G_1 , G_2 , . . . , G_n , b is a G . P } . \]
\[\text{ Let r be the common ratio of this G . P } . \]
\[ \therefore r = \left( \frac{b}{a} \right)^\frac{1}{n + 1} \]
\[\text{ And }, G_1 = ar, G_2 = a r^2 , G_3 = a r^3 , . . . , G_n = a r^n \]
\[\text{ Now, product of n geometric means } = G_1 \cdot G_2 \cdot G_3 \cdot . . . \cdot G_n = \left( ar \right)\left( a r^2 \right)\left( a r^3 \right) . . . \left( a r^n \right)\]
\[ = \left( ar \right)\left( a r^2 \right)\left( a r^3 \right) . . . . . . \left( a r^n \right) \]
\[ = a^n r^{1 + 2 + 3 + . . . + n} \]
\[ = a^n r^\frac{n\left( n + 1 \right)}{2} \]
\[ = a^n \left\{ \left( \frac{b}{a} \right)^\frac{1}{n + 1} \right\}^\frac{n\left( n + 1 \right)}{2} \]
\[ = a^n \left( \frac{b}{a} \right)^\frac{n}{2} \]
\[ = a^\frac{n}{2} b^\frac{n}{2} \]
\[ = \left( ab \right)^\frac{n}{2} \]
\[ \]
\[\]
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