Suppose a, b and c are positive numbers satisfying:

First, we cross multiply for all three equations to get 

a^2 = b, b^2 = 2c, and c^2 = 4a. Thus, we know that a = sqrt(b) from the first equation and b = sqrt(2c) from the second. Next, using the last equation, we find that 

c^2 = 4a ---> c^2 = 4sqrt(b). We already found that b = sqrt(2c), so we can substitute it in to get c^2 = 4*sqrt(sqrt(2c)) = 4*∜2c. Further simplifying it, we get c^8 = 512c ---> c^7 = 512, so c equals the seventh root of 512. 

Lastly, we plug this value of c into the third equation to obtain a = about 1.48599429.

a^2/b  = 1

b^2/c  = 2

c^2/a  = 4

a^2 = b    →   a^4  = b^2

b^2/c →  a^4/c  = 2  →  a^4 = 2c  →  a^4/2  = c  →  a^8/4  = c^2

c^2/a →  (a^8 /4 ) ( 1/a)  =  4

a^7  =  16

a  =  16^(1/7)  =  ⁷√16