# How do you find #3# geometric means between #3# and #1488# ?

##### 1 Answer

#### Explanation:

We are looking for

#color(red)(a)# is the geometric mean of#3# and#color(green)(b)#

#color(green)(b)# is the geometric mean of#color(red)(a)# and#color(blue)(c)#

#color(blue)(c)# is the geometric mean of#color(green)(b)# and#1488#

That will make the following sequence into a geometric one:

#3, color(red)(a), color(green)(b), color(blue)(c), 1488#

If the common ratio is

#1488 = 3 r^4#

So:

#r^4 = 1488/3 = 496 = 2^4*31#

in order that the geometric means be Real and positive, we need to choose the principal

#r = 2root(4)(31)#

Hence

#3*2root(4)(31) = color(red)(6root(4)(31))#

#6root(4)(31)*2root(4)(31) = color(green)(12sqrt(31))#

#12sqrt(31)*2root(4)(31) = color(blue)(24(root(4)(31))^3)#