Write the product of n geometric means between two numbers a and b.

Question:

Write the product of n geometric means between two numbers a and b.

Solution:

Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ geometric means between two quantities $a$ and $b$.

Thus, $a, G_{1}, G_{2}, \ldots, G_{n}, b$ is a G.P.

Let $r$ be the common ratio of this G.P.

$\therefore r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

And, $G_{1}=a r, G_{2}=a r^{2}, G_{3}=a r^{3}, \ldots, G_{n}=a r^{n}$

Now, product of $n$ geometric means $=G_{1} \cdot G_{2} \cdot G_{3} \cdot \ldots \cdot G_{n}=(a r)\left(a r^{2}\right)\left(a r^{3}\right) \ldots\left(a r^{n}\right)$

$=(a r)\left(a r^{2}\right)\left(a r^{3}\right) \ldots \ldots\left(a r^{n}\right)$

$=a^{n} r^{1+2+3+\ldots+n}$

$=a^{n} r^{\frac{n(n+1)}{2}}$

$=a^{n}\left\{\left(\frac{b}{a}\right)^{\frac{1}{n+1}}\right\}^{\frac{n(n+1)}{2}}$

$=a^{n}\left(\frac{b}{a}\right)^{\frac{n}{2}}$

$=a^{\frac{n}{2}} b^{\frac{n}{2}}$

$=(a b)^{\frac{n}{2}}$

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