How do you find 33 geometric means between 33 and 14881488 ?
1 Answer
Explanation:
We are looking for
color(red)(a)a is the geometric mean of33 andcolor(green)(b)b
color(green)(b)b is the geometric mean ofcolor(red)(a)a andcolor(blue)(c)c
color(blue)(c)c is the geometric mean ofcolor(green)(b)b and14881488
That will make the following sequence into a geometric one:
3, color(red)(a), color(green)(b), color(blue)(c), 14883,a,b,c,1488
If the common ratio is
1488 = 3 r^41488=3r4
So:
r^4 = 1488/3 = 496 = 2^4*31r4=14883=496=24⋅31
in order that the geometric means be Real and positive, we need to choose the principal
r = 2root(4)(31)r=24√31
Hence
3*2root(4)(31) = color(red)(6root(4)(31))3⋅24√31=64√31
6root(4)(31)*2root(4)(31) = color(green)(12sqrt(31))64√31⋅24√31=12√31
12sqrt(31)*2root(4)(31) = color(blue)(24(root(4)(31))^3)12√31⋅24√31=24(4√31)3