Problem 11 - A Graphical Approach to Precalculus with Limits

Question

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$-4,-12,-36,-108, \dots$$

Step-by-Step Solution

Answer

The fifth term, \(a_5\), is \(-324\), and the general term is \(a_n = -4 \cdot 3^{n-1}\).

Step 1: Identify the First Term

The first term of the sequence, denoted as \(a_1\), is the first number in the sequence. Here, \(a_1 = -4\).

Step 2: Find the Common Ratio

The common ratio \(r\) of a geometric sequence can be found by dividing the second term by the first term. In this sequence, \(r = \frac{-12}{-4} = 3\).

Step 3: Write the General Formula for the nth Term

The general formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). Substituting the known values: \(a_n = -4 \cdot 3^{n-1}\).

Step 4: Find the Fifth Term

To find \(a_5\), substitute \(n = 5\) into the general formula:\[a_5 = -4 \cdot 3^{5-1} = -4 \cdot 3^4\]Calculate \(3^4 = 81\), so:\[a_5 = -4 \cdot 81 = -324\]

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element that describes how one term in the sequence relates to its previous term. It is consistent across the sequence, and understanding the common ratio allows us to predict and continue the sequence.
To find the common ratio (\(r\)), you simply divide any term in the sequence by the term that comes immediately before it. Using our sequence, we find that dividing the second term, \(-12\), by the first term, \(-4\), gives us:

• \(r = \frac{-12}{-4} = 3\)

This means every term is 3 times the previous term.
Recognizing and calculating the common ratio gives us insight into the nature of the sequence and is foundational for working with geometric sequences.

nth Term Formula

The nth term formula is like a blueprint for finding any term in a geometric sequence without having to list all the previous ones. It provides a shortcut and can be written as:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number you are trying to find.
For our sequence, the first term (\(a_1\)) is \(-4\) and the common ratio (\(r\)) is \(3\).
Plugging these into the formula, we get:

• \(a_n = -4 \cdot 3^{n-1}\)

This formula allows us to calculate any term, such as the fifth term, by replacing \(n\) with 5.
This approach highlights the efficiency of algebraic expressions in revealing the structure of sequences.

Analytic Skills

Analytic skills are the foundation that allows us to dissect and solve mathematical problems like sequences step-by-step. Breaking down a sequence involves recognizing patterns and applying mathematical formulas accurately.
The exercise showcases these skills in a practical setting by requiring the identification of the first term and finding the common ratio correctly before proceeding to formula application.

• Start by closely examining the sequence to determine its pattern.
• Apply the appropriate mathematical procedures to find necessary components like the common ratio or specific terms.
• Ensure to understand each step in isolation to build a coherent understanding of the entire process.

Mastering these skills not only helps with sequences but aids in solving a variety of complex mathematical problems by fostering a methodical approach to problem-solving.