The product of n geometric means between a and b

To find:- Product of n geometric  means between

Let us suppose g1, g2, ............, gn represent geometric means between a and b.

a = a

i.e g1 = ar

 g2 = ar2

gn = arn

b = arn+1

i. e $r^{n+1}=\frac{b}{a}$

$\Rightarrow g_{1} g_{2} g_{3-----} g_{n}=\left(a r^{\cdot}\right)\left(a r^{2}\right)\left(a r^{3}\right)_{---_{-}} a r^{n}$

$=a^{n} r^{1+2+3+_{--+}}$

$=a^{n} r^{\frac{n(n+1)}{2}} \quad\left[\because 1+2+_{---}+n=\frac{n(n+1)}{2}\right]$

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

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

Here $g_{1} g_{2} g_{3----} g_{n}=\left(a b^{\frac{1}{2}}\right)^{n}=(a b)^{\frac{n}{2}}$