Question

If n geometric means be inserted between a and b then prove that their product is $(ab{)}^{n/2}$.

Answer 1

G.P. series of (n+2) terms:

$a,G_1,G_2,..........,G_n,b$

Let r be the common ratio:

$\therefore b=a{r}^{n+1}$

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

G is the single GM between a and b, so we have:

$G=(ab{)}^{\frac{1}{2}}$

The product $G_1\times G_2\times ...............\times G_n$

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

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

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

$=(ab{)}^{\frac{n}{2}}$