Problem 14 - College Algebra

Question

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. $$ a_{1}=8, r=0.3 $$

Step-by-Step Solution

Answer

The first five terms are 8, 2.4, 0.72, 0.216, and 0.0648.

Step 1: Identify the First Term

The first term of the geometric sequence is given as \( a_1 = 8 \). This will be the starting point for identifying the sequence.

Step 2: Identify the Common Ratio

The common ratio for this geometric sequence is given as \( r = 0.3 \). This ratio will be used to find each subsequent term in the sequence.

Step 3: Calculate the Second Term

To find the second term \( a_2 \), multiply the first term by the common ratio: \( a_2 = a_1 \cdot r = 8 \times 0.3 = 2.4 \).

Step 4: Calculate the Third Term

To find the third term \( a_3 \), multiply the second term by the common ratio: \( a_3 = a_2 \cdot r = 2.4 \times 0.3 = 0.72 \).

Step 5: Calculate the Fourth Term

To find the fourth term \( a_4 \), multiply the third term by the common ratio: \( a_4 = a_3 \cdot r = 0.72 \times 0.3 = 0.216 \).

Step 6: Calculate the Fifth Term

To find the fifth term \( a_5 \), multiply the fourth term by the common ratio: \( a_5 = a_4 \cdot r = 0.216 \times 0.3 = 0.0648 \).

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

In a geometric sequence, the **common ratio** is a key element that helps determine the progression of the sequence. It is the constant factor you multiply with to transition from one term to the next.
In simple terms, imagine a magic number that, when multiplied with a term, gives you the next term. This is the common ratio. For example, if we start with an apple (our first term) and multiply it by the common ratio, we get a slice of the apple (our next term).
Using the exercise given, let's say we begin with 8 apples, but our common ratio is 0.3. Each next term would be 30% of the previous term.
The unique feature of a geometric progression is how this ratio remains unchanged across all terms.

• This constancy gives geometric sequences their regular and predictable nature.
• Thus, it forms the backbone of these sequences by shaping the entire pattern.

Understanding this makes it easy to construct the sequence step by step.

Identifying the First Term in a Sequence

The **first term** in a geometric sequence is where you start, denoted by \( a_1 \). Think of it as your starting amount or the first element of the series around which everything evolves.
In the context of the provided exercise, the first term is 8. This term sets the stage for the entire sequence, and every subsequent term is a derivative of this one.
Determining this element is crucial because:

• Without it, you won't have a starting point to apply your common ratio.
• It's the reference from which all positions in the sequence derive their values through multiplication by the common ratio.

It is always given explicitly in the exercise, ensuring clarity in constructing the sequence. Therefore, identifying your first term is the initial step in exploring any geometric sequence.

Exploring the Nature of Mathematical Sequences

A **mathematical sequence** is a structured list of numbers arranged according to a pattern or a rule. There are various types of sequences, but one of the most orderly forms is the geometric sequence.
In these sequences, numbers grow based on a deterministic rule, like multiplying by a common ratio.

• Being able to spot and create sequences is a crucial skill, allowing you to predict future events or solve problems efficiently.
• This methodical arrangement facilitates easier communication of numerical data and relationships in various fields of mathematics and applied sciences.

In geometric patterns, the sequence offers insight into exponential growth or decay, depending on whether the common ratio is greater or smaller than 1. This powerful concept extends beyond just puzzles and homework into practical applications, such as calculating interest rates or population growth, where patterns follow a regular sequence.

The Role of College Algebra in Solving Sequences

**College Algebra** is a foundation-level course designed to equip students with essential algebraic concepts, including the understanding and manipulation of sequences.
It introduces students to not only arithmetic sequences but also to geometric ones, leading them toward solving real-world problems using these concepts.
Throughout this learning experience, students learn about:

• The significance of starting points (first terms)
• Recognizing patterns like common ratios
• Constructing sequences systematically from simple rules

College Algebra provides rigorous training and insights into how mathematical principles underpin complex systems. Mastery of this fundamental topic empowers learners to tackle more challenging mathematical disciplines such as calculus or linear algebra. Understanding sequences through College Algebra develops a critical thinking approach that is vital in both academic and everyday settings.