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In a geometric sequence, each number is called a "term." A geometric sequence is a list of numbers that follow a specific pattern where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This fixed number is known as the "common ratio." To further understand, let's see how terms work in our example:
- The first term, given as \(a_1 = -3\), starts off the sequence.- The second term in the sequence is 30, calculated by multiplying the first term by the common ratio (-10).- This process continues with each subsequent term calculated by multiplying the previous term by that same common ratio.
A sequence could keep going indefinitely, but for practical purposes, we often focus on finding only a few terms. Knowing how the terms are connected through this ratio gives us a clear way to calculate further terms and understand the sequence structure.

The common ratio is a critical factor in a geometric sequence, determining how we get from one term to the next. This ratio remains constant for the entire sequence. Let's break down how the common ratio is used:

• The common ratio, denoted as \(r\), in our example is -10, meaning that each term is obtained by multiplying the previous one by -10.
• This multiplication can lead to negative or positive terms, depending on the sign of the common ratio and the current term.
• A positive common ratio produces terms keeping their original sign, while a negative common ratio alternates the signs of the terms throughout the sequence.

To determine the impact of a common ratio further, consider the sequence alternates between positive and negative values, given that the common ratio in our case is negative. So, understanding this concept not only helps in computing terms but also in visualizing how the sequence behaves.

One essential element in grasping geometric sequences is the sequence formula. This formula is what allows us to compute any term in the sequence without recalculating all the previous terms. The formula for a geometric sequence is:
\[a_n = a_1 \times r^{(n-1)}\] Where:

• \(a_n\) is the \(n\)-th term you want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number.

Using this formula allows you to easily find any term in the sequence. For example, suppose you want the third term, \(a_3\). Substitute into the formula: \(a_3 = -3 \times (-10)^{2} = -300\).This approach saves time and effort when dealing with lengthy sequences and provides a clearer understanding of how each part of the sequence is connected through multiplication and power.