Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ \frac{5}{13}+\frac{10}{11}+\frac{15}{9}+\frac{20}{7} $$

Step-by-Step Solution

Answer

The sum is \( \sum_{n=1}^{4} \frac{5n}{15-2n} \).

Step 1: Look for a Pattern

Identify the pattern in the series. Observe that the numerators are 5, 10, 15, and 20, which can be expressed as multiples of 5. Similarly, the denominators are 13, 11, 9, and 7, which decrease by 2 with each term. Use this information to form a general term formula.

Step 2: Develop a General Term Formula

Express each term in the series as a fraction, \( \frac{5n}{15-2n} \), where \( n \) represents the term's position in the sequence starting from \( n=1 \). Check this formula with the given terms: For \( n=1 \), we get \( \frac{5\times1}{15-2\times1} = \frac{5}{13} \). The pattern holds true for the remaining terms.

Step 3: Write the Summation Notation

Using the general term \( \frac{5n}{15-2n} \), write the series in summation notation. Since there are four terms in the series, \( n \) should range from 1 to 4. Thus, the summation notation is \( \sum_{n=1}^{4} \frac{5n}{15-2n} \).

Key Concepts

Pattern Recognition in Series

When working with a series like \( \frac{5}{13} + \frac{10}{11} + \frac{15}{9} + \frac{20}{7} \), the first step is to search for patterns in both the numerators and the denominators. This makes it easier to express the entire series using a general formula.

• Numerators: In this series, the numerators are 5, 10, 15, and 20. Notice that they are all multiples of 5. Specifically, these are the sequences of 5 times the natural numbers 1, 2, 3, and 4 respectively.
• Denominators: Examine the denominators, which are 13, 11, 9, and 7. This sequence decreases by 2 as you move from one term to the next.

Recognizing these patterns is crucial because it allows us to form a general formula for the series. With the numerators being a constant multiple and the denominators having a consistent pattern, it becomes manageable to express the entire series concisely.

General Term Formula

The general term formula helps encapsulate the entire series into a single expression.
For our series, the pattern in numerators and denominators guides us in formulating this expression. To derive the general term formula, observe:

• The numerators are given by \( 5n \), where \( n \) is the position of the term in the series. For example, \( n=1 \) gives the numerator 5, and \( n=2 \) gives 10.
• The denominators follow the pattern \( 15 - 2n \). At \( n=1 \), the denominator is 13; at \( n=2 \), it's 11, and so on.

Putting it together, the general term for this series is \( \frac{5n}{15-2n} \). This formula represents each term of the series in terms of \( n \), ensuring each follows the identified pattern. Always verify the formula by substituting values of \( n \) to ensure it accurately represents the given terms in the series.

Expressing Series in Summation Notation

Summation notation provides a concise way to represent series using the general term formula. For our series \( \frac{5}{13} + \frac{10}{11} + \frac{15}{9} + \frac{20}{7} \), we have derived the general formula \( \frac{5n}{15-2n} \).
To express this series using summation notation:

• Identify the range of \( n \). In our example, the sequence runs from the 1st to the 4th term, so \( n \) ranges from 1 to 4.
• Use the summation symbol \( \Sigma \) to represent the addition of terms across this range.

Putting these together, the summation notation for the series is: \[\sum_{n=1}^{4} \frac{5n}{15-2n}\]This notation effectively communicates the entire sum of the series using the general term, providing an efficient and tidy form of expression. Understanding and practicing this form of notation simplifies the manipulation of more complex series in the future.