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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

         

Given that the first few terms of a geometric progression are \(6, 18, 54, 162 \ldots \), what is the common ratio of the geometric progression?

Given that the first, second, and third terms of a geometric progression are:

\[3, 9, 27 ... , \]

what is the fifth term?

In the geometric sequence \[\{ a_i \}_{i \geq 1} = \{ -7, 14, -28, \ldots, \} \] what is the value of \(n\) such that \(a_n = 224?\)

If the six numbers \[7, k_1, k_2, k_3, k_4, \frac{7}{32}\] form a geometric sequence in this order, what is \(k_2?\)

Given that the third, fourth, and fifth terms of a geometric progression are:

\[ \ldots, 45, 135, 405, \ldots, \]

what is the second term?

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