Algebra
# Geometric Progressions

If $\{a_n\}$ is a geometric progression with $a_1 = 5$ and $a_5 = 80$, what is the value of $a_3$?

The geometric mean of the geometric sequence ${2, a_2, a_3}$ is 10. What is the value of $a_3?$

**Note.** The geometric mean of three numbers $a_1, a_2, a_3$ is $\sqrt[3]{a_1\cdot a_2 \cdot a_3}.$ For example, the geometric mean of $2, 4,$ and $8$ is $\sqrt[3]{2\cdot(4)\cdot8} = \sqrt[3]{64} = 4.$

$x, y, z^2$ is a geometric progression.

Each of $x, y, z$ are integers.

$x + y + z^2 < 120$.

**Find the largest possible value of $x + y + z^2$.**