Geometric Progressions
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 \(a_3?\)