Question

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric means of those two quantities.

Answer 1

Let $G_1,G_2,G_3,.....G_n,$ be n geometric means between two quantities a and b. Then,

a, $G_1,G_2,G_3,.....G_n$ b is a G.P.

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

Then,

$r={\left(\frac{b}{a}\right)}^{\frac{1}{n+1}}$ and, (G_1 = ar,~G_2 = ar^2,~G_3 = ar^3, ...., G_n = ar^n).

$\therefore G_1,G_2,G_3,.....G_n=\left(ar\right)\left(a{r}^2\right)\left(a{r}^3\right)....\left(a{r}^n\right)$

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

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

$={\left\{\sqrt{ab}\right\}}^n$

$=G^n$

Where $G=\sqrt{ab}$ is the single

geometric means between a and b.