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In a geometric sequence, each term is derived from the previous term by multiplying it with a fixed number known as the common ratio. This ratio is pivotal because it defines the pattern of the sequence.
When dealing with a sequence, once you know any term and the common ratio, you can determine any other term in the sequence. To find the common ratio in the problem, we had two terms: the third term, \(h_3 = \frac{2}{27}\), and the sixth term, \(h_6 = \frac{2}{729}\). By setting up the equations based on these terms and then dividing them, we arrived at \(r^3 = \frac{1}{27}\). This equation tells us how the sequence has grown over three steps. By calculating the cube root of both sides, the common ratio \(r\) is found to be \(\frac{1}{3}\).

• Key idea: The common ratio is constant and allows the sequence to progress in a predictable manner.
• Finding the common ratio helps in finding any term in the sequence.
• In this exercise, identifying \(r\) involved dividing the sixth term by the third term and calculating the cube root of the result.

The general formula for a geometric sequence provides a universal way to calculate any term of that sequence. It is expressed as \(h_n = h_0 \cdot r^n\), where \(h_n\) is the term you want to find, \(h_0\) is the first term, and \(r\) is the common ratio.
This formula is powerful because it connects every term to the very first term, scaling it by multiplying with \(r^n\). In our problem, even though the value of \(h_0\) wasn't given directly, the use of the given terms and the common ratio allowed us to calculate it.

• Importance: This formula is essential for finding any term within the sequence as long as the first term and the common ratio are known.
• Flexibility: Even if you don't know \(h_0\), calculating it using other known terms and the ratio is straightforward, as seen when finding it to be \(\frac{2}{3}\) in this exercise.

The cube root is a specific type of root that undoes cubing a number. For any number \(x\), the cube root is a number \(y\) such that \(y^3 = x\). Understanding cube roots is crucial when the relationship between elements involves cubes.In our exercise, we found ourselves needing to determine the cube root when solving for the common ratio of the sequence. After simplifying the division of given terms, \(r^3\) equates to \(\frac{1}{27}\). To find \(r\), we needed the cube root of \(\frac{1}{27}\), which is \(\frac{1}{3}\).

• Definition: The cube root reverses the cube operation and is represented in mathematics as \(\sqrt[3]{x}\).
• Application: Essential when resolving equations involving terms raised to the third power.
• In geometry and algebra, roots are fundamental for dealing with polynomial equations and sequences.