Question

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 $y$.

Answer 1

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