91Ó°ÊÓ

The common ratio is a key component of any geometric sequence. It defines the factor by which we multiply a term to get the subsequent term. In the above exercise, the **common ratio** is expressed in the recursive formula as \( a_{n} = -6a_{n-1} \). This tells us that each term in the sequence is -6 times the previous term.

• If the common ratio is positive, the sequence's terms all maintain the same sign over consecutive terms.
• If it’s negative, as in our exercise, each term alternates in sign compared to its predecessor.

In our sequence, the common ratio is -6, resulting in the signs of the sequence terms altering with each step, keeping the pattern alternating in terms of positivity and negativity.

Sequence terms are the individual numbers that make up a sequence. In a geometric sequence, we start with an initial term, usually denoted as \( a_1 \), and generate subsequent terms through the multiplication of the common ratio. For the given problem, the initial term \( a_1 \) is -2. We then compute the following terms using the formula, repeatedly applying the common ratio:

• The second term: \( a_2 = -6 \, \times \,-2 = 12 \)
• The third term: \( a_3 = -6 \, \times \, 12 = -72 \)
• The fourth term: \( a_4 = -6 \, \times \, -72 = 432 \)
• The fifth term: \( a_5 = -6 \, \times \, 432 = -2592 \)

Understanding sequence terms in a geometric sequence helps one recognize the rapid growth or decay based on the common ratio value.

A recursive formula defines each term of a sequence using the preceding term(s). In geometric sequences, this often involves multiplying the previous term by the common ratio. In our exercise, the recursive formula given is \( a_{n} = -6 a_{n-1} \):

• \( a_{n} \) refers to the current term we are interested in.
• \( a_{n-1} \) is the term directly preceding it.

With the recursive formula, you need to know just the first term and the rule involving the common ratio to find all subsequent terms. This is different from an explicit formula where terms are calculated directly from their position number in the sequence. Utilizing recursive formulas can seem a bit like following a breadcrumb trail where each step depends on the one before it, making them invaluable for geometric sequences. Recursive relations are a compact way to define sequences, and though sometimes tricky to initiate, they enable easy step-by-step generation of a sequence.