Suppose $x$,$y$, and $z$ form a geometric sequence. if you know that $x+y+z=18$ and $x^2+y^2+z^2=612$, find the value of $y$.

Because x, y, z form a geometric sequence, therefore the common ratio is

r = y/x = z/y

That is,

y² = xz (1)

We are given:

x + y + z = 18

Therefore

x + z = 18 - y (2)

Also,

x² + y² + z² = 612

Therefore, from (1), obtain

x² + z² + xz = 612

(x + z) ² - xz = 612

From (1) and (2), obtain

(18 - y) ² - y² = 612

324 - 36y + y² - y² = 612

-36y = 288

y = - 8

Answer: y = - 8