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In a geometric series, finding the nth term is a common task. The nth term formula is a very useful tool for this calculation. It is essential to understand that the nth term formula for a geometric series can be expressed as: \[ a_n = a_1 \times r^{(n-1)} \]where:- \( a_n \) is the term in the sequence that we wish to find,- \( a_1 \) is the first term of the series,- \( r \) is the common ratio between terms,- \( n \) is the term number or position in the sequence.The formula helps in pinpointing the exact term in a sequence without listing all preceding terms. By understanding each element in this formula, especially how changes in the common ratio or the first term affect the series, students can better grasp how geometric sequences expand or contract. This formula lays the foundational understanding for working with geometric sequences.

The common ratio \( r \) is a crucial part of any geometric series. In a geometric sequence, the common ratio is the factor by which each term is multiplied to get the next one. It remains constant throughout the sequence.To find the common ratio, divide any term by the previous term. For instance, if you have two consecutive terms from a geometric sequence, say \( a_2 \) and \( a_1 \), the common ratio \( r \) would be:\[ r = \frac{a_2}{a_1} \]In our example problem, the common ratio \( r \) was given as \(-2\), indicating that each term is obtained by multiplying the previous term by \(-2\). This means the signs of the terms will alternate, and each term will either increase or decrease in magnitude based on this multiplier.Understanding the common ratio helps identify the pattern of the series and predicts future terms.

Calculating specific terms in a geometric sequence involves substituting values into the nth term formula. Let’s break it down:1. **Identify Known Values:** Determine the first term \( a_1 \) and the common ratio \( r \). For example, \( a_1 = -4 \) and \( r = -2 \) in the provided exercise.2. **Substitute into Formula:** Use these known values in the nth term formula: \[ a_n = -4 \times (-2)^{(n-1)} \]3. **Simplification:** Depending on the term you need (), calculate \((-2)^{n-1}\) and multiply the result by \(-4\).By following these steps, you can quickly identify any term in the sequence without listing each one. This process is particularly helpful when dealing with large sequences or when calculating terms farther along in the series.