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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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