Question

Insert five geometric means between 486 and 2/3.What is the inverse of ratio of consecutive terms ?

Answer 1

Let the G.P be $486,x_1,x_2,x_3,x_4,x_5,\frac{2}{3}$ of 7 terms

$\frac{2}{3}=T_7=a{r}^6=\left(486\right)r^6$

$r^6=\frac{1}{729}=\frac{1}{3^6}$

$r=\pm \frac{1}{3}$ Take $r=\frac{1}{3}$
$x_1=T_2=ar=486.\left(\frac{1}{3}\right)=162$
$x_2=T_3=a{r}^2=486\left(\frac{1}{9}\right)=54$
$x_3=18,x_4=6,x_5=2$
Hence the five geometric means are $162,54,18,6,2$