Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$4,-6,9,-13.5, \ldots$$

Step-by-Step Solution

Answer

The nth term is \(4 \cdot \left(-\frac{3}{2}\right)^{n-1}\), the fifth term is 20.25, and the eighth term is -68.4375.

Step 1: Identify the First Term and Common Ratio

The first term of the sequence, denoted as \(a\), is the first number in the sequence, which is \(4\). The common ratio \(r\) is obtained by dividing the second term by the first term: \(-6 \div 4 = -\frac{3}{2}\). Verify this ratio with subsequent terms like \(9 \div (-6) = -\frac{3}{2}\) and \(-13.5 \div 9 = -\frac{3}{2}\).

Step 2: Write the General Formula for the nth Term

For a geometric sequence, the general term \( a_n \) is given by \( a_n = a \cdot r^{n-1} \). Substitute \(a = 4\) and \(r = -\frac{3}{2}\) into this formula to get \( a_n = 4 \cdot \left(-\frac{3}{2}\right)^{n-1} \).

Step 3: Calculate the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the formula: \( a_5 = 4 \cdot \left(-\frac{3}{2}\right)^{4} \). Calculate \( \left(-\frac{3}{2}\right)^4 = \left(\frac{9}{4}\right)^2 = \frac{81}{16} \). Therefore, \( a_5 = 4 \times \frac{81}{16} = \frac{81}{4} = 20.25 \).

Step 4: Calculate the Eighth Term

To find the eighth term \(a_8\), substitute \(n = 8\): \( a_8 = 4 \cdot \left(-\frac{3}{2}\right)^{7} \). Calculate \( \left(-\frac{3}{2}\right)^7 = -\left(\frac{2187}{128}\right) \). Thus, \( a_8 = 4 \times \left(-\frac{2187}{128}\right) = -\frac{8748}{128} = -68.4375 \).

Key Concepts

nth term calculation

In a geometric sequence, calculating the nth term is a fundamental step. Understanding this helps you pinpoint any term in the sequence without listing all previous terms. The formula for finding the nth term, denoted as \( a_n \), depends on two key components: the first term \( a \) and the common ratio \( r \). The general formula used is:

• \( a_n = a \cdot r^{n-1} \)

This formula takes the first term and multiplies it by the common ratio raised to the power of \( (n-1) \). To use this effectively, it’s important to correctly identify both \( a \) and \( r \). Ensure precision when substituting into the formula, especially with negative ratios and higher power calculations. This approach significantly streamlines finding specific terms in any geometric sequence.

common ratio

The common ratio is a crucial element in every geometric sequence. It indicates how each term is related to the previous one. You find it by dividing any term by the one before it.In our example sequence 4, -6, 9, -13.5, ..., the ratio \( r \) was found by calculating \( -6 \div 4 = -\frac{3}{2} \). It is essential to verify this ratio by dividing additional terms, such as \( 9 \div (-6) = -\frac{3}{2} \), ensuring consistency throughout the sequence.Key points in identifying the common ratio:

• Make sure calculations adhere to the order of terms.
• Confirm with multiple consecutive terms to avoid errors.

Recognizing and sticking with the right common ratio helps accurately predict and calculate specific terms using the general formula.

general term formula

The core tool for understanding geometric sequences is the general term formula. This formula, \( a_n = a \cdot r^{n-1} \), serves as a blueprint for constructing the sequence. It enables the prediction of any term's value without needing previous terms, provided the first term \( a \) and common ratio \( r \) are known. ### Model Example:Consider our sample sequence:

• First Term \( a = 4 \)
• Common Ratio \( r = -\frac{3}{2} \)

Substitute these into the formula: - \( a_n = 4 \cdot \left(-\frac{3}{2}\right)^{n-1} \)This equation builds the framework for determining any term's value and is crucial for both finding specific terms and analyzing the sequence's properties.

sequence analysis

Sequence analysis in geometric sequences involves understanding patterns and behavior beyond individual terms. By analyzing the terms, we can infer general properties of the sequence. When examining the sequence 4, -6, 9, -13.5, ..., start with both the common ratio \( -\frac{3}{2} \) and the pattern it creates:

• Notice the alternating signs of consecutive terms.
• Examine how the magnitude changes based on the ratio to understand growth or decay behavior.

Factors like whether the ratio is greater or less than zero influence the direction and behavior of the terms:- Negative ratios cause terms to alternate signs.- Absolute values illustrate whether the sequence expands or contracts.Understanding these elements is pivotal for grasping how geometric sequences operate and what conclusions might be drawn from their structure.