Suppose , and are positive numbers satisfying:
a^2/b = 1
b^2/c = 2
c^2/a = 4
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.