Problem 5 - Applied Calculus

Question

Find the sum, if it exists. $$ 100+100(0.85)+100(0.85)^{2}+\cdots+100(0.85)^{10} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 535.42.

Step 1: Identify the Problem Type

The given series \(100 + 100(0.85) + 100(0.85)^2 + \cdots + 100(0.85)^{10}\) is a geometric series. Each term has a common ratio of \(0.85\) and there are 11 terms (from the 0th to the 10th power of \(0.85\)). Our task is to find the sum of this series.

Step 2: Recognize the Geometric Series Formula

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

Step 3: Substitute Known Values

For this series:- The first term \(a = 100\)- The common ratio \(r = 0.85\)- The number of terms \(n = 11\)Substitute these values into the formula:\[ S_{11} = 100 \frac{1-(0.85)^{11}}{1-0.85} \]

Step 4: Calculate Powers of the Common Ratio

Compute \(0.85^{11}\).Using a calculator, you find:\[ 0.85^{11} \approx 0.1968744040 \]

Step 5: Compute the Sum

Now substitute the calculated value back into the sum formula:\[ S_{11} = 100 \frac{1-0.1968744040}{0.15} \]Perform the calculation:\[ S_{11} = 100 \frac{0.803125596}{0.15} \approx 100 \times 5.35417064 \approx 535.42 \]

Step 6: Conclude with the Result

The sum of the series is approximately \(535.42\).

Key Concepts

Series Sum Calculation

To determine the sum of a geometric series, the whole process involves identifying key values and substituting them into a specific formula. In this particular problem, the function to find the series' sum revolves around calculating repeatedly multiplying numbers that follow a clear pattern. The series in question is a typical example of this kind.
Here's what you'll typically need to do:

• First, identify the first term of the series, referred to as \(a\). For the current series, \(a\) is 100.
• Second, determine the common ratio, \(r\). For this exercise, the common ratio is 0.85.
• Count the number of terms, denoted by \(n\). For this given sequence, there are 11 terms (from \(0\) to \(10\) power).

With these three parts, you can plug them into the geometric series formula and calculate the sum. It’s crucial to ensure every step follows the exact order to properly calculate parts like \(r^n\) and the final division to get the sum.

Common Ratio

The common ratio in a geometric series is simply the factor by which we multiply each term to get the next term in the sequence. For this series, the common ratio, denoted as \(r\), is 0.85.
Understanding the common ratio is vital because it helps us see the consistent pattern in the series. It’s what differs a geometric series from other sequences. A few points to note about the common ratio:

• If \(r = 1\), the series doesn't change with each term, making it constant and not quite geometric.
• If \(r\) is between 0 and 1, as in this exercise, the terms decrease progressively, illustrating a "shrinking" series.
• If \(r > 1\), the terms would grow with each progression, turning into an "expanding" series.

Accurately identifying the common ratio lets us use the geometric series formula effectively, ensuring we can calculate the series sum correctly.

Geometric Series Formula

The geometric series formula is a powerful tool to calculate the sum of terms that have a defined multiplicative pattern, known as the common ratio. The formula for finding the sum \(S_n\) of the first \(n\) terms is:\[ S_n = a \frac{1-r^n}{1-r} \]In this formula:

• \(a\) is the first term of the series.
• \(r\) represents the common ratio.
• \(n\) is the total number of terms.

Let’s apply this to our series. Plugging in our values—\(a = 100\), \(r = 0.85\), and \(n = 11\)—into the formula gives:\[ S_{11} = 100 \frac{1-(0.85)^{11}}{1-0.85} \]By computing \(0.85^{11}\), then substituting everything back into the equation, you can solve to find the sum.
The geometric series formula simplifies the problem significantly, providing a structured way to handle various similar problems you might face.