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A geometric sequence is a collection of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In simple terms, you start with your first term and get each subsequent term by multiplying by this common ratio repeatedly.

For example, given the sequence formula, like the one in the exercise, \( a_n = \frac{(-1)^{n}}{3^{n}} \), although this formula includes an alternating pattern (which we'll touch upon later), it also has the structure of a geometric sequence.

Here's why:

• All terms involve powers of 3 in the denominator, which affects the geometric rate. In this case, each denominator is a power of 3, showing how the sequence value reduces geometrically.

Even though we have another factor (\((-1)^n\)), the growing power of 3 plays a critical role in the sequence's geometric nature. The pattern of 3 as the base of the denominator ensures the sequence's size shrinks by a ratio of \(\frac{1}{3}\) for each additional term.

An alternating sequence is a type of sequence where the signs of its terms switch between positive and negative consistently. In our exercise example, the term \((-1)^{n}\) is present, which causes the alternation.

Here's how it works:

• When \(n\) is an odd number (like 1 or 3), \((-1)^n\) becomes \(-1\).
• When \(n\) is an even number (like 2 or 4), \((-1)^n\) becomes \(1\).

This alternating pattern results in terms flipping back and forth between negative and positive. This particular concept allows sequences to model phenomena with recurring opposing changes, such as alternating current in electricity or swings around a balance point.

Each number in a sequence is referred to as a sequence term. When handling sequences, especially in exercises like the one given, understanding and identifying terms clearly is essential. Each term in a sequence has an order, starting from the first term, second term, and so forth, defined usually by \(n\).

Here’s a breakdown to help with sequences:

• Terms are calculated in sequence by using the given formula with the term number in place of \(n\).
• In the exercise, for \(n = 1\), the term is \(a_1 = \frac{-1}{3}\); for \(n = 2\), it is \(a_2 = \frac{1}{9}\); and so on.

Identifying and calculating every term correctly involves plugging the term position into the sequence formula. Each answer provides the position and value, building up the series progressively. Recognizing each term's placement and pattern contributes to a deeper grasp of the sequence itself.