Problem 49 - Precalculus Mathematics for Calculus

Question

Find the sum. $$\sum_{k=0}^{10} 3\left(\frac{1}{2}\right)^{k}$$

Step-by-Step Solution

Answer

The sum is \( \frac{6141}{1024} \).

Step 1: Identify the First Term

We start by identifying the first term of the series given by the expression inside the summation.At \( k = 0 \), the first term is \( 3\left(\frac{1}{2}\right)^{0} = 3 \times 1 = 3 \). So, the first term \( a = 3 \).

Step 2: Identify the Common Ratio

The common ratio \( r \) in the geometric series is the factor by which each term is multiplied to get the next term.From the expression \( 3\left(\frac{1}{2}\right)^{k} \), the common ratio is \( \frac{1}{2} \) because each subsequent term involves multiplying by \( \frac{1}{2} \).

Step 3: Identify the Number of Terms

The sum is taken from \( k=0 \) to \( k=10 \), meaning the sum encompasses 11 terms. Thus, the number of terms \( n = 11 \).

Step 4: Use the Geometric Series Formula

The formula for the sum of a geometric series is:\[ S_n = a \frac{1-r^n}{1-r} \]where \( a = 3 \), \( r = \frac{1}{2} \), and \( n = 11 \). Substitute these values into the formula.

Step 5: Calculate the Power of the Common Ratio

Compute \( r^n \):Given \( r = \frac{1}{2} \) and \( n = 11 \), \[ \left(\frac{1}{2}\right)^{11} = \frac{1}{2048} \].

Step 6: Substitute and Simplify

Substitute \( a = 3 \), \( r = \frac{1}{2} \), and \( (\frac{1}{2})^{11} = \frac{1}{2048} \) into the geometric series formula:\[ S_{11} = 3 \frac{1 - \frac{1}{2048}}{1 - \frac{1}{2}} = 3 \frac{2047/2048}{1/2} = 3 \times \frac{2047}{1024} \].

Step 7: Final Calculation

Continue simplifying:\[ S_{11} = 3 \times \frac{2047}{1024} = \frac{6141}{1024} \].You can further divide by 3:\[ S_{11} \approx 5.996 \].However, the fraction \( \frac{6141}{1024} \) is already in simplest form.

Key Concepts

Sum of series

In mathematics, when dealing with a geometric series, the sum is found by adding together all the terms in the sequence. A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. To find the sum of a fixed number of terms in a geometric series, we use the geometric series formula.

For the series \( \sum_{k=0}^{10} 3\left(\frac{1}{2}\right)^{k} \), the task was to determine its sum up to the 11th term (since we start from \( k=0 \), we include 11 terms in total).

Using the formula:

• \( S_n = a \frac{1-r^n}{1-r} \)

It's crucial to correctly identify the first term \( a \), the common ratio \( r \), and the total number of terms \( n \) to effectively apply this formula.

Common ratio

The common ratio is vital in identifying how the terms of a geometric sequence progress. It is the constant factor by which each term of the sequence is multiplied to produce the next term.

In our example, \( 3\left(\frac{1}{2}\right)^{k} \), the common ratio \( r \) can be easily spotted within the expression as \( \frac{1}{2} \). Each term is found by taking the previous term and multiplying it by \( \frac{1}{2} \).

This consistent multiplication factor explains why the series is called "geometric"—each subsequent term grows (or shrinks) by this fixed ratio.

Geometric sequence

A geometric sequence, also known as a geometric progression, forms the basis of a geometric series. It consists of numbers where each term is the product of the previous term and the common ratio.

In this problem, \( 3, 1.5, 0.75, \ldots \), represents our geometric sequence when evaluated at \( k=0, 1, 2, \ldots \).

Each number follows this multiplication rule by the common ratio \( \frac{1}{2} \).

Recognizing the pattern and rule of multiplication helps us see the underlying structure of the series, which is crucial for solving related problems.

Series formula

The series formula is a mathematical tool used to compute the sum of the terms in a geometric series.

The general form of the formula we use to find the sum of the first \( n \) terms is:

• \( S_n = a \frac{1-r^n}{1-r} \)

This formula requires knowledge of:

• \( a \): the first term of the series.
• \( r \): the common ratio between consecutive terms.
• \( n \): the total number of terms to be added.

Inserting these values into the formula yields the sum. Understanding this formula is crucial since it unlocks the ability to quickly sum any geometric series without manually adding each term.