Algebra

Geometric Progressions

Geometric Progressions Warmup

         

If {an}\{a_n\} is a geometric progression with a1=5a_1 = 5 and a5=80a_5 = 80, what is the value of a3a_3?

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If a1,3,a3,12,a5a_1, 3, a_3, 12, a_5 is a geometric progression, which of the following is a possible value of a5?a_5?

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The geometric mean of the geometric sequence 2,a2,a3{2, a_2, a_3} is 10. What is the value of a3?a_3?

Note. The geometric mean of three numbers a1,a2,a3a_1, a_2, a_3 is a1⋅a2⋅a33.\sqrt[3]{a_1\cdot a_2 \cdot a_3}. For example, the geometric mean of 2,4,2, 4, and 88 is 2⋅(4)⋅83=643=4.\sqrt[3]{2\cdot(4)\cdot8} = \sqrt[3]{64} = 4.

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  • x,y,z2 x, y, z^2 is a geometric progression.

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

  • x+y+z2<120x + y + z^2 < 120.

  • Find the largest possible value of x+y+z2x + y + z^2.

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If a1,5,a3,45,a5a_1, 5, a_3, 45, a_5 is a geometric progression, which of the following is a possible value of a3?a_3?

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