Problem 11 - Applied Calculus

Question

Find the sum, if it exists. $$ 65+\frac{65}{1.02}+\frac{65}{(1.02)^{2}}+\cdots+\frac{65}{(1.02)^{18}} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 1013.91.

Step 1: Identify the Series Type

The given series is a geometric series because each term is obtained by multiplying the previous term by a constant ratio. The formula for a geometric series is \( a + ar + ar^2 + \cdots + ar^{n-1} \). In our series, \( a = 65 \) and the common ratio \( r = \frac{1}{1.02} \approx 0.9804 \). There are 18 terms in the series.

Step 2: Use the Geometric Series Sum Formula

The sum \( S_n \) of the first \( n \) terms of a geometric series is given by \( S_n = \frac{a(1-r^n)}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Substitute \( a = 65 \), \( r = 0.9804 \), and \( n = 18 \) into the formula.

Step 3: Calculate the Value of \( r^{n} \)

Calculate \( r^{n} = (0.9804)^{18} \). This requires using a calculator to get a precise value for \( (0.9804)^{18} \). You should find that \( (0.9804)^{18} \approx 0.6947 \).

Step 4: Substitute Values into the Formula

Substitute the values into the geometric series formula: \( S_{18} = \frac{65(1 - 0.6947)}{1 - 0.9804} \). This simplifies to \( S_{18} = \frac{65 \times 0.3053}{0.0196} \).

Step 5: Calculate the Sum

Now, compute \( S_{18} = \frac{65 \times 0.3053}{0.0196} \). This results in \( S_{18} \approx 1013.91 \).

Key Concepts

Sum of Series

A geometric series is a sum of terms where each term is a specific multiple, called a "common ratio," of the previous term. In our exercise, the task is to compute the sum of the series given by:
\[65 + \frac{65}{1.02} + \frac{65}{(1.02)^{2}} + \cdots + \frac{65}{(1.02)^{18}}\]Understanding how to find the sum is crucial for evaluating such series.
To find the sum of a geometric series, we use the sum formula:

• First Term \( (a) \)
• Common Ratio \( (r) \)
• Number of Terms \( (n) \)

The sum formula for a geometric series is:\[S_n = \frac{a(1-r^n)}{1-r}\] This formula enables us to calculate the sum accurately when we know the first term, the common ratio, and the number of terms.
Applying this formula to our series helps us find that:\[S_{18} \approx 1013.91\]

Geometric Sequence Formula

The geometric sequence formula is a fundamental tool to understand geometric series. Each term in a geometric sequence is obtained by multiplying the previous term by a fixed constant, known as the "common ratio". The general formula for the \(n^{th}\) term is given by:
\[a_n = ar^{n-1}\]where:

• \(a\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the position of the term in the sequence.

For example, in our exercise, using \(a = 65\) and \(r = \frac{1}{1.02}\), you can express any term in the sequence using this formula, enabling a clear understanding of how the sequence grows.

Common Ratio

The common ratio \((r)\) in a geometric series is the factor you multiply by to get from one term to the next. Identifying the common ratio is key to solving problems involving geometric series.
In the given series:\[r = \frac{1}{1.02} \approx 0.9804\]Note how each term is obtained by multiplying the previous term by 0.9804.

• When the common ratio \(|r| < 1\), terms decrease as they move forward, creating a shrinking series.
• If \(|r| > 1\), terms grow larger, expanding as the series progresses.

For our series, the common ratio is less than one, meaning the terms gradually decrease in size, converging towards zero.

Finite Series

A finite series in mathematics is a series that ends after a certain number of terms, unlike an infinite series. In solving such series, the concept is more straightforward because you only deal with a set number of terms.
The given exercise involves 18 terms, making it a finite geometric series. It makes evaluating sums feasible by using formulas and computations.
Knowing the number of terms \((n)\) is essential when applying the geometric series sum formula:\[S_n = \frac{a(1-r^n)}{1-r}\]The presence of a finite number of terms means that calculations can be completed, resulting in a precise sum of the series, as seen in the result of \(S_{18} \approx 1013.91\). This method provides a definite answer, unlike infinite series, which may require limits to evaluate.