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# Geometric Progressions

A geometric progression is a sequence of numbers where the previous term is multiplied by a constant to get the next term. 1, 2, 4, 8,... is a geometric sequence where each term is multiplied by 2.

# Geometric Progressions Warmup

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

If $a_1, 3, a_3, 12, a_5$ is a geometric progression, which of the following is a possible value of $$a_5?$$

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

If $a_1, 5, a_3, 45, a_5$ is a geometric progression, which of the following is a possible value of $$a_3?$$

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