Problem 8 - Precalculus

Question

Write the first five terms of each geometric sequence. $$a_{n}=-6 a_{n-1}, \quad a_{1}=-2$$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are -2, 12, -72, 432, -2592.

Step 1: Identify the Initial Value and Ratio

First, let's note the key components of the formula given. The sequence starts with \(a_{1}=-2\), this is the first term. It also includes a common ratio indicated by the formula \(a_{n}=-6 a_{n-1}\), which implies each term is -6 times the term before it.

Step 2: Calculate the Second Term

Using the formula, the second term of the sequence can be calculated as \(a_{2}= -6 a_{1} = -6 (-2) = 12\). Thus, our sequence now is -2, 12.

Step 3: Calculate the Third Term

Continuing onto the next term, according to our formula this would be \(a_{3}= -6 a_{2}= -6 (12) = -72\). Thus, our sequence is -2, 12, -72.

Step 4: Calculate the Fourth Term

Using the same formula, calculate the fourth term: \(a_{4}= -6 a_{3}= -6 (-72) = 432\). The sequence is now -2, 12, -72, 432.

Step 5: Calculate the Fifth Term

Calculate the fifth term in the same way: \(a_{5}= -6 a_{4}= -6(432) = -2592\). Now, the five term sequence is -2, 12, -72, 432, -2592.

Key Concepts

Common Ratio

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

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.

Recursive Formula

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.