Problem 40 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the finite geometric series. The first 15 terms of the series defined by \(a_{n}=2\left(\frac{1}{3}\right)^{n}\)

Step-by-Step Solution

Answer

The sum of the first 15 terms is approximately 1.

Step 1: Identify Constants

First, define the initial term \( a \) and the common ratio \( r \) of the geometric series. From the given sequence \( a_n = 2\left(\frac{1}{3}\right)^{n} \), we have that \( a = 2 \cdot \left(\frac{1}{3}\right)^1 = \frac{2}{3} \) and the common ratio \( r = \frac{1}{3} \).

Step 2: Write the Sum Formula for Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated using the formula \[ S_n = a \frac{1 - r^n}{1 - r} \]. Here, \( a \) is the first term and \( r \) is the common ratio.

Step 3: Substitute Values into the Formula

Substitute \( a = \frac{2}{3} \), \( r = \frac{1}{3} \), and \( n = 15 \) into the sum formula: \[ S_{15} = \frac{2}{3} \times \frac{1 - \left(\frac{1}{3}\right)^{15}}{1 - \frac{1}{3}} \].

Step 4: Calculate \( r^n \)

Compute \( \left(\frac{1}{3}\right)^{15} \). A calculator can help for this step: \( \left(\frac{1}{3}\right)^{15} \approx 3.0518 \times 10^{-8} \).

Step 5: Calculate Denominator and Numerator Separately

First, calculate the denominator: \( 1 - \frac{1}{3} = \frac{2}{3} \). Now, calculate the numerator: \( 1 - \left(\frac{1}{3}\right)^{15} \approx 1 - 3.0518 \times 10^{-8} \approx 0.9999999695 \).

Step 6: Calculate the Sum

Using the previous results, compute the sum: \[ S_{15} = \frac{2}{3} \times \frac{0.9999999695}{\frac{2}{3}} = 0.9999999695 \], which says that the sum is almost 1.

Key Concepts

Geometric Sequence

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
This means if you have a first term, let's call it \( a \), and a common ratio \( r \), every term in the sequence can be described as:

• \( a \) (the first term)
• \( ar \) (the second term)
• \( ar^2 \) (the third term)
• \( ar^n \) (the \( n+1 \)th term)

The magic of a geometric sequence lies in its simplicity and consistency—every term follows directly from the one before it by multiplying by the common ratio.

Sum of Series Formula

The sum of a geometric series is the total when you add up terms of a geometric sequence. For a finite geometric series, where you only consider a certain number of terms, there's a handy formula:
\[S_n = a \frac{1-r^n}{1-r}\]Here, \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
This formula arises from manipulating the expressions for each sequence term, leveraging how each term is a product with the common ratio.
It's important because it gives a simple way to calculate the total of many series, without having to add each term manually.

Common Ratio

In a geometric sequence, the common ratio is the constant factor between consecutive terms. It determines the sequence's growth pattern.
For example, if the common ratio \( r \) is greater than 1, the sequence expands, creating larger numbers with each step. Conversely, if \( r \) is between 0 and 1, the sequence contracts, moving closer to zero.

• A common ratio of 1 results in a constant sequence, without change.
• A negative common ratio will alternate the terms' signs.

Understanding the common ratio is key to predicting how the series behaves and solving related mathematical problems.

Sequence Terms

Understanding the terms of a sequence is crucial for working with series and their sums. In our geometric sequence, the terms are generated using the first term \( a \) and the common ratio \( r \).
To find any term in the sequence, you can use the formula:
\(a_n = a \cdot r^{n-1}\)
where \( n \) is the term number you're interested in.
This formula shows that each term builds steadily upon the previous one, guided by the consistent multiplication with the common ratio.

• The first term is simply \( a \).
• The subsequent terms change based on multiplying the previous term by \( r \).

Visualizing a geometric series in this way helps make sense of how the entire sequence progresses.