Problem 77 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=0}^{15}2\left(\dfrac{4}{3}\right)^n \)

Step-by-Step Solution

Answer

The sum of the given geometric series is \( S_{16} = -6 \cdot (1 - (\frac{4}{3})^{16})\)

Step 1: Identify the first term, common ratio and number of terms

We first identify the parameters of the geometric series. In terms of the general expression \(a \cdot r^n\), where the number of terms sums from n=0 to n=15, our first term \(a\) is 2, the common ratio \(r\) is \(\frac{4}{3}\), and the number of terms is \(n=16\).

Step 2: Use the formula to find the sum

We substitute the identified values into the formula for the sum of a geometric series given by \[S_n=a \left ( \dfrac{1-r^n}{1-r} \right )\]. Substituting \( a = 2\), \( r = \frac{4}{3}\) and \( n = 16 \) into the formula, we get \( S_{16} = 2 \cdot \left(\frac{1 - (\frac{4}{3})^{16}}{1 - \frac{4}{3}}\right)\). The sum is found by evaluating this expression.

Step 3: Simplify and evaluate the expression

The expression simplifies to \( S_{16} = 2 \cdot \left(\frac{1 - (\frac{4}{3})^{16}}{-\frac{1}{3}}\right)\) and further simplifies to \( S_{16} = -6 \cdot (1 - (\frac{4}{3})^{16})\). Evaluating this expression gives the sum of the geometric series.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a critical component that defines the relationship between consecutive terms. Simply put, the common ratio is the factor by which we multiply one term to get the next term in the sequence.
For instance, if you have a sequence of numbers like 2, 6, 18, 54, you can identify the common ratio by dividing any term by the previous term. Here, each term is multiplied by 3 to get the next term, meaning the common ratio is 3.
In the context of the exercise, the sequence under consideration has a common ratio of \( \frac{4}{3} \). This indicates that each term is \( \frac{4}{3} \) times the preceding term. Identifying the common ratio is pivotal because it is used in calculating the sum of the geometric series and helps us understand the progression pattern of the sequence elements.

Finite Series Sum

The sum of a finite geometric series is a concept where we aim to add up a set number of terms in a geometric sequence. While an infinite geometric series can continue indefinitely, a finite series has a specific end point, making its sum easier to compute.
To calculate the sum of a finite geometric series, you use a specific formula: \[ S_n = a \left( \frac{1 - r^n}{1 - r} \right) \]where:

• \( S_n \) is the sum of the series
• \( a \) is the first term of the series
• \( r \) is the common ratio
• \( n \) is the number of terms

In the exercise presented, you substitute \( a = 2 \), \( r = \frac{4}{3} \), and \( n = 16 \) to find the series sum. By correctly applying these values into the formula, you derive the result for the sum. Understanding this process and formula is essential for anyone dealing with geometric series, providing a straightforward strategy to sum finite sequences without calculating each term individually.

Geometric Sequence Formula

The geometric sequence formula is a fundamental expression that describes any geometric sequence. This formula allows us to determine any term in the sequence given a specific position.
A geometric sequence is expressed as \( a_n = a \cdot r^{n-1} \), where:

• \( a_n \) is the \( n^{th} \) term
• \( a \) is the first term
• \( r \) is the common ratio
• \( n \) is the term number

This formula provides an efficient method to retrieve any term without listing all preceding terms. For example, if you have a series starting at 2 with a common ratio of \( \frac{4}{3} \), and you want to find the 5th term, substitute into the formula to get \( 2 \cdot \left(\frac{4}{3}\right)^4 \).
Being able to use this formula effectively is beneficial in various mathematical and real world contexts, as it elegantly encapsulates the characteristics of a geometric progression.