Problem 42 - College Algebra

Question

For the following exercises, find the specified term for the geometric sequence given. Let \(a_{1}=4, a_{n}=-3 a_{n-1} .\) Find \(a_{8}\)

Step-by-Step Solution

Answer

The 8th term, \( a_8 \), is \(-8748\).

Step 1: Understand the Formula of the Geometric Sequence

In this problem, a geometric sequence is given with the first term \( a_1 = 4 \) and a recursive relation \( a_n = -3a_{n-1} \). This tells us that each term is obtained by multiplying the previous term by \(-3\).

Step 2: Calculate the First Few Terms

Let's calculate the first few terms to recognize the pattern. \( a_2 = -3 \times a_1 = -3 \times 4 = -12 \). Next, \( a_3 = -3 \times a_2 = -3 \times (-12) = 36 \). Similarly, \( a_4 = -3 \times a_3 = -3 \times 36 = -108 \).

Step 3: Continue the Pattern Up to a_8

Continue the process: \( a_5 = -3 \times a_4 = -3 \times (-108) = 324 \); \( a_6 = -3 \times a_5 = -3 \times 324 = -972 \); \( a_7 = -3 \times a_6 = -3 \times (-972) = 2916 \); Finally, \( a_8 = -3 \times a_7 = -3 \times 2916 = -8748 \).

Step 4: Verify the Pattern

Each term is obtained by multiplying the previous term by \(-3\), which aligns with the recursive relation. Checking \( a_1 \) through \( a_8 \), each follows the formula \( a_n = -3 a_{n-1} \).

Key Concepts

Recursive Formulas

Recursive formulas provide a way to describe a sequence by using the relationship between any term and its previous term. In a geometric sequence, this relationship is often expressed through a multiplier, called the common ratio. For example, the sequence provided in the exercise follows the recursive formula: - \(a_n = -3a_{n-1}\) This tells us how to generate the next term from the previous one, constantly using \(-3\) as the multiplier.

• Starting Point: Also known as the initial term, here, it's \(a_1 = 4\).
• Multiplier: The value \(-3\) illustrates how each term derives from its predecessor.

Understanding a recursive formula is crucial, as it lays out the direct method of constructing the sequence step-by-step. This recursive pattern helps identify both the nature of the sequence and how it expands or contracts through successive terms.

Algebraic Patterns

Geometric sequences exhibit distinct algebraic patterns, which can be identified once you understand the recursive formula provided. Recognizing these patterns allows for predictions about non-explicit terms without direct calculation.- The recursive formula \(a_n = -3a_{n-1}\) is an algebraic representation. This shows a geometric pattern where every term is the product of the preceding term and a constant factor of \(-3\).

• The sign of terms will alternate due to the negative multiplier.
• The magnitude of terms grows with higher absolute values of each term as it progresses.

Patterns such as alternating signs and growing magnitudes help quickly verify whether we applied the recursive formula correctly. Recognizing these features simplifies following and predicting the sequence's continuation.

Term Calculation

Calculating specific terms in a geometric sequence involves closely following the recursive formula provided.Let's use the example of calculating the eighth term, \(a_8\), which can be achieved by following these steps: 1. **Start with the Initial Term**: Begin with \(a_1 = 4\).2. **Apply the Recursive Formula**: Use \(-3\) as our constant multiplier repeatedly to find subsequent terms: - \(a_2 = -3 \times 4 = -12\) - \(a_3 = -3 \times (-12) = 36\) - Proceed through \(a_4\) to \(a_8\) using the same method.Calculations for terms:- Continue multiplying until the desired term is reached, maintaining careful checks to ensure accuracy in signs and values.Through systematic calculation, following the recursive pattern to the eighth term results in \(a_8 = -8748\).Understanding these steps emphasizes the efficiency of recursive relations in calculating terms and the importance of each calculation's accuracy to maintaining the integrity of the sequence.