Problem 59 - Algebra 2

Question

Write a geometric series for which \(r=\frac{1}{2}\) and \(n=4\)

Step-by-Step Solution

Answer

The geometric series is \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\).

Step 1: Understanding the Components of a Geometric Series

A geometric series is the sum of the terms of a geometric sequence. In a geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \(r\). The total number of terms is denoted \(n\).

Step 2: Identifying Given Values

We are given the common ratio \(r = \frac{1}{2}\) and the number of terms \(n = 4\). We need to construct a geometric series using these values.

Step 3: Writing the Formula for the Terms of the Series

The general term of a geometric series can be written as \(a, ar, ar^2, ar^3, \ldots, ar^{n-1}\) where \(a\) is the first term. Here, \(r = \frac{1}{2}\) and \(n = 4\).

Step 4: Choosing the First Term

To write a specific series, we can choose any non-zero value for the first term \(a\). Let's choose \(a = 1\) for simplicity.

Step 5: Calculating Each Term in the Series

Using \(a = 1\) and \(r = \frac{1}{2}\), we calculate each term:- First term: \(1 \cdot (\frac{1}{2})^0 = 1\)- Second term: \(1 \cdot (\frac{1}{2})^1 = \frac{1}{2}\)- Third term: \(1 \cdot (\frac{1}{2})^2 = \frac{1}{4}\)- Fourth term: \(1 \cdot (\frac{1}{2})^3 = \frac{1}{8}\)

Step 6: Writing the Final Geometric Series

The geometric series for \(n = 4\) with \(r = \frac{1}{2}\) and \(a = 1\) is:\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\).

Key Concepts

Common Ratio

In the context of geometric sequences and series, the **common ratio** is a crucial element. It is the fixed, non-zero number that each term in the geometric sequence is multiplied by to produce the next term. In simpler terms, think of the common ratio as the constant factor that **determines the rate** of increase or decrease in the sequence's terms.
For instance, if the common ratio is less than 1, the terms will progressively decrease, as each subsequent term becomes a fraction of the previous one. Conversely, if the common ratio is greater than 1, the terms will increase in size.
In our example, the common ratio is given as \( r = \frac{1}{2} \). This means each term in the sequence is multiplied by \( 1/2 \) to derive the next. This leads to terms that halve as the sequence progresses. Thus, the sequence decreases in value, showcasing how a common ratio impacts the entire geometric series.

Geometric Sequence

A **geometric sequence** is a number sequence formed by multiplying each term by a fixed number, known as the common ratio, to get the next term. It can be represented generally as:

• First term: \( a \)
• Second term: \( ar \)
• Third term: \( ar^2 \)
• Fourth term: \( ar^3 \)

This pattern continues up to \( n \) terms, where \( n \) is the number of terms. The formula for the \( n^{th} \) term is \( ar^{n-1} \).
In our case, with \( r = \frac{1}{2} \) and \( n = 4 \), we first select a convenient \( a \) value, as it defines the series uniquely. By selecting \( a = 1 \), the sequence becomes \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \). Notice how each term is derived by multiplying by \( \frac{1}{2} \). This hallmark makes identifying and working with geometric sequences efficient and predictable.

Arithmetic Sequence

Unlike a geometric sequence, an **arithmetic sequence** constructs each of its terms by **adding** a constant difference rather than multiplying by a ratio. If you think of the arithmetic sequence as taking steps, each step is equal and consistent.
This is fundamentally different from our geometric sequence, where each term changes based on multiplication. Hence, even though both sequences have a structured formula, they differ in approach:

• **Arithmetic Sequence**: Terms are calculated using addition or subtraction.
• **Geometric Sequence**: Terms are calculated using multiplication or division.

The formula for an arithmetic sequence is \( a_n = a + (n-1) \cdot d \), where \( d \) is the common difference. Comparatively, in our original exercise, we determined a geometric series and highlighted the process of multiplication—not addition—to progress through terms. Understanding these differences aids in distinguishing between sequence types, allowing targeted problem-solving strategies when needed.