Determine the numbers a, b if 3 ,a ,b , 24 is a geometric sequence.

We'll use the theorem of geometric mean of a geometric
sequence.:

a^2 = 3b (1)

b^2
= 24a (2)

We'll raise to square
(1):

a^4 = 9b^2

We'll divide
by 9 both sides:

b^2 = a^4/9
(3)

We'll substitute (3) in
(2):

a^4/9 = 24a

We'll divide
by a:

a^3/9 = 24

We'll cross
multiply and we'll get:

a^3 =
24*9

a^3 = 2^3*3^3

a =
2*3

a =
6

For a = 6, we'll get
b:

b^2 = 24a

b = sqrt
24*6

b =
sqrt144

b =
12

So, for a=6 and b=12, the
terms of the geometric series, whose common ratio is r  =2, are: 3 , 6 , 12 , 24,
....