Find the missing terms of each geometric sequence. (Hint:...

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.)
$12.5, \square, \square, \square, 5.12, \dots$

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Key Concepts

Common Ratio

The common ratio is the fixed multiplier used to progress from one term to the next in a geometric sequence. It is a critical element in determining the sequence’s terms because once the first term and the ratio are known, every term can be computed by repeated multiplication.

Geometric Mean

The geometric mean is a type of average where the nth root of the product of n numbers is taken, and in geometric sequences it manifests as the middle term in a sequence of an odd number of terms. Particularly, the geometric mean property, where the middle term is the geometric mean of the first and last terms, can be utilized to compute unknown terms in the sequence.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant value known as the common ratio. Understanding this structure is key to finding missing terms, as the relationship between any two terms is defined by the power of the common ratio that connects them.

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