Problem 74 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{i=1}^{12}16\left(\dfrac{1}{2}\right)^{i - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is \(32 \times (1 - (0.5)^{12})\).

Step 1: Identify First Element, Common Ratio, Total Elements

First, we identify the first element of the series \(a\), the common ratio \(r\), and the total number of terms \(n\). After examining the series \(16(*(0.5)^{i - 1})\) from \(i = 1\) to \(12\), we see that \(a = 16\), \(r = 0.5\), and \(n = 12\).

Step 2: Apply the Finite Geometric Series formula

Next, we apply the given values for \(a\), \(r\), and \(n\) into the finite geometric series formula. This gives us \(S = 16 * (1 - (0.5)^{12}) / (1 - 0.5)\)

Step 3: Calculate the fraction part

Now, calculate the fraction part of the formula: \((1 - (0.5)^{12}) / (1 - 0.5)\). This equals \(2 \times (1 - (0.5)^{12})\).

Step 4: Solve for S

Finally, we solve for \(S\) by multiplying \(16\) with the result of the fraction we computed in the previous step. The final answer S is \(S = 16 \times 2 \times (1 - (0.5)^{12})\).

Key Concepts

Geometric Sequence

In mathematics, a geometric sequence is simply a series of numbers, each of which is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence follows the pattern:

• \(a, ar, ar^2, ar^3, \, \ldots\)

where

• \(a\)
• \(r\)
• are both constants.

In our original exercise, the sequence starts with \(16\), and each subsequent term is obtained by multiplying the previous term by \(0.5\). This specific pattern and repetition are what define it as a geometric sequence. Geometric sequences are widely applicable in different areas such as finance, sciences, and engineering due to their powerful nature of growth or decay.

Sum of Series

Calculating the sum of a geometric series involves adding up a finite or infinite set of its terms. For a finite geometric sequence, which is the focus of our exercise, we use a specific formula to determine the sum of its terms. The formula for the sum of a finite geometric sequence is: \[ S_n = a \frac{1-r^n}{1-r} \]where:

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

In our particular example, this formula helps us find the total sum of the series from the first term up to the twelfth term. Understanding how to sum these series is crucial when the series represents real-world processes such as population growth or interest compounding, where knowing the total is just as important as understanding individual terms.

Common Ratio

The common ratio is a vital component of a geometric sequence. It is the factor by which we multiply each term of the sequence to get the next term. To find it, you divide any term by the term preceding it. In our example, the common ratio $r$ is $0.5$, indicating that each subsequent term is half of its previous term. This concept is not just crucial in defining a geometric sequence but also in determining the behavior of the sequence as we move along: whether it increases, decreases, or remains constant. A common ratio well-grounded between $0$ and $1$ results in a decreasing sequence, often seen in processes modeling exponential decay.

First Term

The first term, typically denoted as $a$, is the initial element from which a geometric sequence begins. It's essential because not only does it set the starting point of the sequence, but it also influences how large or small the entire sequence is. In our exercise, the first term is $16$, which is significant as it establishes the base value from which all subsequent terms and the series as a whole are derived. Understanding the first term is crucial for accurately applying the formula for the sum of the series, as it directly affects the calculated sum. In real-world situations, the first term could represent an initial investment or the starting population size, and even minute changes to this figure can dramatically impact future projections and calculations.