Problem 84 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=0}^{25}8\left(-\dfrac{1}{2}\right)^i \)

Step-by-Step Solution

Answer

The sum of the geometric sequence is 15.9999999 or approximately 16.

Step 1: Identify the elements

The elements of the series are: first term \(a = 8\), common ratio \(r = -\frac{1}{2}\), and number of terms \(n = 26\).

Step 2: Apply the Formula for the Geometric Sum

The formula is \(S = a \cdot \frac{1-r^n}{1-r}\). Substitute \(a\), \(r\), and \(n\), into the formula which gives \(S = 8 \cdot \frac{1- (-\frac{1}{2})^{26}}{1-(-\frac{1}{2})}\).

Step 3: Simplify the Expression

First, calculate the value of \((- \frac{1}{2})^{26}\), which is \(1.490116e-08\). Then, substitute this into the formula which gives \(S = 8 \cdot \frac{1-1.490116e-08}{1+ \frac{1}{2}}\). After that, perform the operations in the numerator and normalize the denominator, yielding the final result.

Key Concepts

Geometric Series Formula

When working with a finite geometric sequence, a central component is the geometric series formula. This formula helps us find the sum of all terms in the sequence. For a geometric sequence, we start with an initial term, known as the first term, denoted as \(a\). Each subsequent term is obtained by multiplying the previous one by a constant called the common ratio \(r\). When you know these elements, the geometric series formula is as follows:\[S_n = a \cdot \frac{1 - r^n}{1 - r}\]In this formula, \(S_n\) is the sum of the series, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Make sure you understand the components before diving into calculations.

• The numerator, \(1 - r^n\), accounts for the power of the common ratio up to the last term.
• The denominator, \(1 - r\), normalizes the sequence given the geometric progression properties.

This formula is applicable when \(req 1\). If the common ratio is 1, each term of the series is equal, and the sum can be directly obtained by multiplying the number of terms \(n\) by the first term \(a\). Always verify the ratio before applying the formula.

Common Ratio in Series

In a geometric sequence, the common ratio \(r\) plays a critical role because it defines how the sequence progresses. It is the constant factor that each term is multiplied by to get to the next one. For example, if your sequence starts with 8 and the common ratio is \(-\frac{1}{2}\), this means that every next term will be half the previous term in magnitude but with alternating signs.

• To determine the common ratio, divide any term by its preceding term.
• The sign and magnitude of \(r\) affect the sequence's direction and growth.

Understanding the common ratio is key for predictions and calculations in a sequence. A positive ratio implies all terms are of the same sign, whereas a negative ratio results in alternating signs.Knowing \(r\) helps predict whether the sequence terms will grow, decrease or oscillate. As \(r\) is raised to higher powers in the sequence, the impact on each term may cause significant changes, crucial for evaluating the sum in a geometric series.

Sum of Series

When tasked with finding the sum of a finite geometric sequence, you are essentially looking to compute the entire series collectively. This is where logic meets practicality; you use formulas to work with more complex series efficiently.The sum of a geometric series can be derived when parameters like first term \(a\), common ratio \(r\), and number of terms \(n\) are known. With our formula \(S_n = a \cdot \frac{1 - r^n}{1 - r}\), you substitute these values to get the result.Let's break down the process:

• First, determine the first term \(a\), which is the original starting term of the sequence.
• Identify the common ratio \(r\), as you will elevate this value to the power \(n\) (number of terms) in the numerator of the formula.
• The denominator \(1-r\) allows adjustment for the repeated multiplication of terms.

By simplifying the expression using these components, you efficiently calculate the total sum. Always perform a check on your steps to ensure all calculations are executed seamlessly and you correctly interpret each value. This makes handling geometric series systematic and less prone to error.